Download Trig: Sine, Cosine, and Tangent 215

Trig: Sine, Cosine, and Tangent 215
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Class Outline
Class Outline
Objectives
Right Triangle Relationships
Labeling Right Triangles
Sine, Cosine, and Tangent
SOHCAHTOA
Sine: Finding a Missing Dimension
Cosine: Finding a Missing Dimension
Tangent: Finding a Missing Dimension
Finding a Missing Angle
Cosecant, Secant, and Cotangent
Calculating Tapers
Taper per Foot: Sample Problem
Taper per Foot: Solution
Finding a Taper Angle: Sample Problem
Finding a Taper Angle: Sample Problem #2
Sample Problem #2: Solution
Summary
Lesson: 1/17
Objectives
l Describe the relationship between the sides and angles of a right triangle.
l Identify the sides of a right triangle according to its reference angle.
l List the most common trig ratios.
l Explain the phrase SOHCAHTOA.
l Use the sine ratio to solve for a missing dimension.
l Use the cosine ratio to solve for a missing dimension.
l Use the tangent ratio to solve for a missing dimension.
l Solve for a missing angle using a trigonometric ratio.
l Use a less common trig ratio to solve for a missing dimension.
l Describe common methods for specifying tapers in blueprints.
l Find the dimensions of a right triangle formed by a conical taper.
l Calculate the taper per foot of a conical taper.
l Solve for the total included angle of a conical taper.
Figure 1. Trigonometric ratios are used to find
the total included angle of a taper in a shop
print.
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Figure 2. A calculator allows you to easily find
the values for sine, cosine, and tangent.
Lesson: 1/17
Objectives
l Describe the relationship between the sides and angles of a right triangle.
l Identify the sides of a right triangle according to its reference angle.
l List the most common trig ratios.
l Explain the phrase SOHCAHTOA.
l Use the sine ratio to solve for a missing dimension.
l Use the cosine ratio to solve for a missing dimension.
l Use the tangent ratio to solve for a missing dimension.
l Solve for a missing angle using a trigonometric ratio.
l Use a less common trig ratio to solve for a missing dimension.
l Describe common methods for specifying tapers in blueprints.
l Find the dimensions of a right triangle formed by a conical taper.
l Calculate the taper per foot of a conical taper.
l Solve for the total included angle of a conical taper.
Figure 1. Trigonometric ratios are used to find
the total included angle of a taper in a shop
print.
Figure 2. A calculator allows you to easily find
the values for sine, cosine, and tangent.
Lesson: 2/17
Right Triangle Relationships
A right triangle is one of the most common shapes found in shop drawings. A right triangle is a
triangle with a 90° angle and two acute angles, like those shown in Figure 1. Like any triangle, a right triangle has three sides and three angles.
You can use the sides of a right triangle to find the measurement of its interior angles. Likewise,
you can also use the measurement of its angles to find information about its sides. However, you
must know at least one of two possible combinations to find this information. You must have the
measurement of a side and any angle besides the 90° angle, or you must have the measurements of any two sides, as you can see in Figure 2.
The angles and sides of a right triangle establish trigonometric ratios. You can use these ratios in
the shop to solve for the missing sides and angles in a print. The most common ratios include sine,
cosine, and tangent.
Figure 1. A right triangle has a 90° angle and In this class, you will learn how to use sine, cosine, and tangent to find information about the sides two acute angles.
Copyright
© of
2015
Tooling
U, LLC.
All will
Rights
Reserved.
and
angles
right
triangles.
You
also
learn how to use these ratios to find information in
sample shop prints.
Lesson: 2/17
Right Triangle Relationships
A right triangle is one of the most common shapes found in shop drawings. A right triangle is a
triangle with a 90° angle and two acute angles, like those shown in Figure 1. Like any triangle, a right triangle has three sides and three angles.
You can use the sides of a right triangle to find the measurement of its interior angles. Likewise,
you can also use the measurement of its angles to find information about its sides. However, you
must know at least one of two possible combinations to find this information. You must have the
measurement of a side and any angle besides the 90° angle, or you must have the measurements of any two sides, as you can see in Figure 2.
The angles and sides of a right triangle establish trigonometric ratios. You can use these ratios in
the shop to solve for the missing sides and angles in a print. The most common ratios include sine,
cosine, and tangent.
Figure 1. A right triangle has a 90° angle and In this class, you will learn how to use sine, cosine, and tangent to find information about the sides two acute angles.
and angles of right triangles. You will also learn how to use these ratios to find information in
sample shop prints.
Figure 2. To find information for right triangles,
you must know the measurement of an angle
and one side or the measurement of two sides.
Lesson: 3/17
Labeling Right Triangles
Before you can find the measurements for the sides and angles of a right triangle, you must
understand how to label these sides and angles using the trig ratios. First, the reference angle is
your measured, known angle. However, the reference angle must be an angle other than the 90° angle that determines the name of each side.
For example, consider the right triangle in Figure 1. Side "a" across from the 58° reference angle is referred to as the opposite side. Side "b," which is next to the reference angle, is referred to as
the adjacent side. Side "c" is the longest side and is always across from the 90° angle. This side is
known as the hypotenuse.
Every right triangle has two possible reference angles. Keep in mind that the adjacent and opposite
sides change depending on which reference angle you use, as Figure 2 shows. A side that is
opposite of one acute angle is adjacent to the other acute angle. It depends which angle you use as
Figure 1. The sides of a right triangle are
the reference angle. However, the hypotenuse is always the longest side of the triangle and
labeled according to its reference angle.
opposite the 90° angle.
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Lesson: 3/17
Labeling Right Triangles
Before you can find the measurements for the sides and angles of a right triangle, you must
understand how to label these sides and angles using the trig ratios. First, the reference angle is
your measured, known angle. However, the reference angle must be an angle other than the 90° angle that determines the name of each side.
For example, consider the right triangle in Figure 1. Side "a" across from the 58° reference angle is referred to as the opposite side. Side "b," which is next to the reference angle, is referred to as
the adjacent side. Side "c" is the longest side and is always across from the 90° angle. This side is
known as the hypotenuse.
Every right triangle has two possible reference angles. Keep in mind that the adjacent and opposite
sides change depending on which reference angle you use, as Figure 2 shows. A side that is
opposite of one acute angle is adjacent to the other acute angle. It depends which angle you use as
Figure 1. The sides of a right triangle are
the reference angle. However, the hypotenuse is always the longest side of the triangle and
labeled according to its reference angle.
opposite the 90° angle.
Figure 2. The adjacent and opposite sides of a
right triangle change depending on which
reference angle you use.
Lesson: 4/17
Sine, Cosine, and Tangent
Trigonometric ratios allow you to find missing angles and dimensions on a right triangle. Figure 1
illustrates the three most common ratios:
l
l
l
The sine of an angle is the side opposite the angle divided by the hypotenuse.
The cosine of an angle is the side next to the angle, or the adjacent side, divided by the
hypotenuse.
The tangent of an angle is the side opposite the angle divided by the side adjacent to the
angle.
Remember, you use these ratios when you know either the reference angle and one side, or the
values for two sides. For example, if you have a given reference angle, a given hypotenuse, and you
must find the side opposite the reference angle, you use the sine ratio. Also, in most cases, the
value you are solving for is on top of the ratio, and the side on the bottom is the value you know.
Sine, cosine, and tangent are often long numbers with many decimal places. Consequently, these
numbers are often rounded to the nearest decimal place. In addition, these ratios are often looked
up in tables, or a calculator is used to quickly find the values. On a calculator, sine is written as SIN, cosine
is COS,
and
tangent
is TAN,
as you
can see in Figure 2.
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© 2015
Tooling
U, LLC.
All Rights
Reserved.
Figure 1. Trig ratios allow you to find the
missing angles and dimensions on a right
triangle.
Lesson: 4/17
Sine, Cosine, and Tangent
Trigonometric ratios allow you to find missing angles and dimensions on a right triangle. Figure 1
illustrates the three most common ratios:
l
l
l
The sine of an angle is the side opposite the angle divided by the hypotenuse.
The cosine of an angle is the side next to the angle, or the adjacent side, divided by the
hypotenuse.
The tangent of an angle is the side opposite the angle divided by the side adjacent to the
angle.
Remember, you use these ratios when you know either the reference angle and one side, or the
values for two sides. For example, if you have a given reference angle, a given hypotenuse, and you
must find the side opposite the reference angle, you use the sine ratio. Also, in most cases, the
value you are solving for is on top of the ratio, and the side on the bottom is the value you know.
Sine, cosine, and tangent are often long numbers with many decimal places. Consequently, these
numbers are often rounded to the nearest decimal place. In addition, these ratios are often looked
up in tables, or a calculator is used to quickly find the values. On a calculator, sine is written as SIN, cosine is COS, and tangent is TAN, as you can see in Figure 2.
Figure 1. Trig ratios allow you to find the
missing angles and dimensions on a right
triangle.
Figure 2. A calculator is used to quickly find the
values of sine, cosine, and tangent.
Lesson: 5/17
SOHCAHTOA
Whenever you know certain values on a right triangle, you must determine which trig ratio you use
to solve for missing sides or angles. However, remembering which ratio to choose can be difficult. If
you are given the hypotenuse and are looking for the opposite side of your angle, you must use
the sine ratio. If you are given the hypotenuse and a side adjacent to your angle, you must use the
cosine ratio. Finally, if you are given both the opposite and adjacent sides of your angle, you must
use the tangent ratio.
There are a variety of different methods that help you remember these relationships. However, one
of the most common is the word SOHCAHTOA. As you can see in Figure 1, each letter in the word
represents these ratios in order: "S" for sine, followed by "O" for opposite, and "H" for hypotenuse.
Next, you have "C" for cosine, "A" for adjacent, and "H" for hypotenuse. Finally, you have "T" for
tangent, "O" for opposite, and "A" for adjacent. Remembering the word SOHCAHTOA is a simple
method that allows you to visualize the relationship between the trig ratios.
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Figure 1. SOHCAHTOA is a simple method that
allows you to visualize the relationship
between the trig ratios.
Lesson: 5/17
SOHCAHTOA
Whenever you know certain values on a right triangle, you must determine which trig ratio you use
to solve for missing sides or angles. However, remembering which ratio to choose can be difficult. If
you are given the hypotenuse and are looking for the opposite side of your angle, you must use
the sine ratio. If you are given the hypotenuse and a side adjacent to your angle, you must use the
cosine ratio. Finally, if you are given both the opposite and adjacent sides of your angle, you must
use the tangent ratio.
There are a variety of different methods that help you remember these relationships. However, one
of the most common is the word SOHCAHTOA. As you can see in Figure 1, each letter in the word
represents these ratios in order: "S" for sine, followed by "O" for opposite, and "H" for hypotenuse.
Next, you have "C" for cosine, "A" for adjacent, and "H" for hypotenuse. Finally, you have "T" for
tangent, "O" for opposite, and "A" for adjacent. Remembering the word SOHCAHTOA is a simple
method that allows you to visualize the relationship between the trig ratios.
Figure 1. SOHCAHTOA is a simple method that
allows you to visualize the relationship
between the trig ratios.
Lesson: 6/17
Sine: Finding a Missing Dimension
Consider the right triangle in Figure 1. You have a reference angle of 27°, and the hypotenuse has a 2.2361 length. The side of the triangle that you must find is "a," which is located opposite the
27° reference angle. Remember, placing the side that you are looking for on top of the equation makes the equation easier to solve. In this case, you are looking for the opposite side (O) over the
hypotenuse (H), which is given. If you think of SOHCAHTOA, you know that you must use the sine
ratio.
Your equation will look like this: SIN 27° = a / 2.2361, as shown in Figure 2. To find the sine of 27°, use your calculator. Enter the number 27, and press the SIN key. If you round off the number
in your display, you find that 0.4540 is the sine of 27 degrees. Your equation will now read 0.4540
= a / 2.2361. Next, multiply both sides of the equation by the hypotenuse (0.4540 x 2.2361 =
1.0152). You find that a = 1.0152.
Figure 1. You must use the sine ratio to solve
for the missing dimension.
Figure 2. Using the sine ratio, you find that a =
1.0152 in length.
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Lesson: 6/17
Sine: Finding a Missing Dimension
Consider the right triangle in Figure 1. You have a reference angle of 27°, and the hypotenuse has a 2.2361 length. The side of the triangle that you must find is "a," which is located opposite the
27° reference angle. Remember, placing the side that you are looking for on top of the equation makes the equation easier to solve. In this case, you are looking for the opposite side (O) over the
hypotenuse (H), which is given. If you think of SOHCAHTOA, you know that you must use the sine
ratio.
Your equation will look like this: SIN 27° = a / 2.2361, as shown in Figure 2. To find the sine of 27°, use your calculator. Enter the number 27, and press the SIN key. If you round off the number
in your display, you find that 0.4540 is the sine of 27 degrees. Your equation will now read 0.4540
= a / 2.2361. Next, multiply both sides of the equation by the hypotenuse (0.4540 x 2.2361 =
1.0152). You find that a = 1.0152.
Figure 1. You must use the sine ratio to solve
for the missing dimension.
Figure 2. Using the sine ratio, you find that a =
1.0152 in length.
Lesson: 7/17
Cosine: Finding a Missing Dimension
The steps for using other trig ratios are very similar. Consider the right triangle in Figure 1. You
have a reference angle of 39°, and the hypotenuse is 3.2169 in length. The side of the triangle that
you must find is "b," which is located adjacent to the 39° reference angle. In this case, you are looking for the adjacent side (A) over the hypotenuse (H). If you use SOCAHTOA, you know that
you must use the cosine ratio.
Your equation will look like this: COS 39° = b / 3.2169, as shown in Figure 2. Using your calculator,
enter the number 39, and press the COS key. Your cosine is 0.7771 after rounding the number in
your display. Your equation now reads 0.7771 = b / 3.2169. Next, multiply both sides of the
equation by the hypotenuse (0.7771 x 3.2169 = 2.5000). You find that b = 2.5000.
Figure 1. You must use the cosine ratio to
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solve for the missing dimension.
Lesson: 7/17
Cosine: Finding a Missing Dimension
The steps for using other trig ratios are very similar. Consider the right triangle in Figure 1. You
have a reference angle of 39°, and the hypotenuse is 3.2169 in length. The side of the triangle that
you must find is "b," which is located adjacent to the 39° reference angle. In this case, you are looking for the adjacent side (A) over the hypotenuse (H). If you use SOCAHTOA, you know that
you must use the cosine ratio.
Your equation will look like this: COS 39° = b / 3.2169, as shown in Figure 2. Using your calculator,
enter the number 39, and press the COS key. Your cosine is 0.7771 after rounding the number in
your display. Your equation now reads 0.7771 = b / 3.2169. Next, multiply both sides of the
equation by the hypotenuse (0.7771 x 3.2169 = 2.5000). You find that b = 2.5000.
Figure 1. You must use the cosine ratio to
solve for the missing dimension.
Figure 2. Using the cosine ratio, you find that b
= 2.5000 in length.
Lesson: 8/17
Tangent: Finding a Missing Dimension
Some problems may not directly involve the hypotenuse. Consider the right triangle in Figure 1.
You have a reference angle of 48°, and the adjacent side of the reference angle is 1.5000 in length.
The side of the triangle that you must find is "c," which is located opposite the 48° angle. In this case, you are looking for the opposite side (O) over the adjacent side (A). If you use SOHCAHTOA,
you find that you must use the tangent ratio.
Your equation will look like this: Tan 48° = c / 1.5000, as shown in Figure 2. Enter the number 48, and press the TAN key. Your display should read 1.1106 after rounding. Your equation now reads
1.1106 = c / 1.5000. Next, multiply 1.1106 by the adjacent side (1.1106 x 1.5000 = 1.6659). You
find that c = 1.6659.
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Figure 1. You must use the tangent ratio to
Lesson: 8/17
Tangent: Finding a Missing Dimension
Some problems may not directly involve the hypotenuse. Consider the right triangle in Figure 1.
You have a reference angle of 48°, and the adjacent side of the reference angle is 1.5000 in length.
The side of the triangle that you must find is "c," which is located opposite the 48° angle. In this case, you are looking for the opposite side (O) over the adjacent side (A). If you use SOHCAHTOA,
you find that you must use the tangent ratio.
Your equation will look like this: Tan 48° = c / 1.5000, as shown in Figure 2. Enter the number 48, and press the TAN key. Your display should read 1.1106 after rounding. Your equation now reads
1.1106 = c / 1.5000. Next, multiply 1.1106 by the adjacent side (1.1106 x 1.5000 = 1.6659). You
find that c = 1.6659.
Figure 1. You must use the tangent ratio to
solve for the missing dimension.
Figure 2. Using the tangent ratio, you find that
c = 1.6659 in length.
Lesson: 9/17
Finding a Missing Angle
With some right triangles, you are given the values for two sides, but you must change the
equation to find the measurement of the reference angle. In the right triangle in Figure 1, you are
given the value of the hypotenuse (9.9002) and the value of the side opposite the reference angle
(6.8730). Remember, if you are using the opposite side and the hypotenuse, you must use the
sine ratio.
Your equation will look like this: SIN X = 6.8730 / 9.9002, and "X" represents your unknown angle,
as you can see in Figure 2. Using your calculator, enter the numbers 6.8730, press the divide sign
(÷), and enter 9.9002. Press the equal sign (=), and your display should read 0.6942. Your equation now reads SIN X = 0.6942.
There is one more step to this problem. Remember that you must find angle X in degrees. To do
this, press the 2nd key on your calculator and the SIN key. In the 2nd mode, sine is expressed as
SIN to the negative one power, or SIN -1. This function "works backward" from the ratio to find your
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2015 Tooling
U, is
LLC.
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answer
in©degrees,
which
43.9657
degrees.
If you must express your answer in degrees,
minutes, and seconds, for some calculators, you can press the 2nd key again and the equal sign
(=). Your display will now read 43° 57' 56".
Lesson: 9/17
Finding a Missing Angle
With some right triangles, you are given the values for two sides, but you must change the
equation to find the measurement of the reference angle. In the right triangle in Figure 1, you are
given the value of the hypotenuse (9.9002) and the value of the side opposite the reference angle
(6.8730). Remember, if you are using the opposite side and the hypotenuse, you must use the
sine ratio.
Your equation will look like this: SIN X = 6.8730 / 9.9002, and "X" represents your unknown angle,
as you can see in Figure 2. Using your calculator, enter the numbers 6.8730, press the divide sign
(÷), and enter 9.9002. Press the equal sign (=), and your display should read 0.6942. Your equation now reads SIN X = 0.6942.
There is one more step to this problem. Remember that you must find angle X in degrees. To do
this, press the 2nd key on your calculator and the SIN key. In the 2nd mode, sine is expressed as
SIN to the negative one power, or SIN -1. This function "works backward" from the ratio to find your
answer in degrees, which is 43.9657 degrees. If you must express your answer in degrees,
minutes, and seconds, for some calculators, you can press the 2nd key again and the equal sign
(=). Your display will now read 43° 57' 56".
Keep in mind that you can "work backward" using any of the three trig ratios to find an unknown
angle in degrees. The steps are generally the same, as long as you use the correct trig ratio that
matches the specific sides and reference angle in your problem.
Figure 1. You must use the sine ratio to solve
for the missing angle.
Figure 2. Using the sine ratio, you find that
angle X is slightly less than 44 degrees.
Lesson: 10/17
Cosecant, Secant, and Cotangent
In addition to sine, cosine, and tangent, Figure 1 illustrates the three less common trig ratios that
may be used:
l
l
l
Cosecant is the reverse of the sine ratio. In other words, sine is the opposite side over the
hypotenuse, and cosecant is the hypotenuse over the opposite side.
Secant is the reverse of the cosine ratio. In other words, cosine is the adjacent side over the
hypotenuse, and secant is the hypotenuse over the adjacent side.
Cotangent is the reverse of the tangent ratio. In other words, tangent is the opposite side
over the adjacent side, and cotangent is the adjacent side over the opposite side.
When finding these measurements using your calculator, there is an additional step required. For
example, the right triangle in Figure 2 has an 18° reference angle, and you are given the length of Copyright
2015opposite
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LLC.You
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0.9748
for© the
side.
must find
the hypotenuse. Remember that placing the unknown
value on top of the equation makes the equation easier to solve.
Figure 1. Cosecant, secant, and cotangent are
the reverse of the sine, cosine, and tangent
ratios respectively.
Lesson: 10/17
Cosecant, Secant, and Cotangent
In addition to sine, cosine, and tangent, Figure 1 illustrates the three less common trig ratios that
may be used:
l
l
l
Cosecant is the reverse of the sine ratio. In other words, sine is the opposite side over the
hypotenuse, and cosecant is the hypotenuse over the opposite side.
Secant is the reverse of the cosine ratio. In other words, cosine is the adjacent side over the
hypotenuse, and secant is the hypotenuse over the adjacent side.
Cotangent is the reverse of the tangent ratio. In other words, tangent is the opposite side
over the adjacent side, and cotangent is the adjacent side over the opposite side.
When finding these measurements using your calculator, there is an additional step required. For
example, the right triangle in Figure 2 has an 18° reference angle, and you are given the length of 0.9748 for the opposite side. You must find the hypotenuse. Remember that placing the unknown
value on top of the equation makes the equation easier to solve.
Figure 1. Cosecant, secant, and cotangent are
the reverse of the sine, cosine, and tangent
ratios respectively.
"Hypotenuse over opposite" means that you must use the cosecant ratio. Your equation will look
like this: CSC 18° = c / 0.9748, as Figure 3 shows. Enter the number 18 and press the SIN key. Your display should read 0.3090. Remember, you must press SIN because it is the reverse of
cosecant, and there is not a button for CSC. Next, press the 1/X key. This tells the calculator to
take the opposite of SIN. Your display should now read 3.2361. Multiply this number by the
opposite side (3.2361 X 0.9748 = 3.1545) to find that the hypotenuse c equals 3.1545 in length.
Figure 2. You can use the cosecant ratio to
solve for the missing dimension in this problem.
Figure 3. Using the cosecant ratio, you find
that hypotenuse c = 3.1545 in length.
Lesson: 11/17
Calculating Tapers
A round taper is a gradual decrease in the diameter of a part from one end to another. The
amount of taper in any given length of a part is found by subtracting the size of the small end from
the large end. On a blueprint, a taper forms a right triangle, with the angle of the taper as the
reference angle and the tapered edge as the hypotenuse.
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The difference in diameter along 1 foot of a part’s length is called the taper per foot (TPF). Tapers
are usually expressed as a ratio and are calculated in inches per foot. For example, 3.5:12 is a
Lesson: 11/17
Calculating Tapers
A round taper is a gradual decrease in the diameter of a part from one end to another. The
amount of taper in any given length of a part is found by subtracting the size of the small end from
the large end. On a blueprint, a taper forms a right triangle, with the angle of the taper as the
reference angle and the tapered edge as the hypotenuse.
The difference in diameter along 1 foot of a part’s length is called the taper per foot (TPF). Tapers
are usually expressed as a ratio and are calculated in inches per foot. For example, 3.5:12 is a
common ratio, which indicates a decrease of 3.5 inches in diameter for every foot in length. To
calculate a taper, first determine the length of the taper. Next, determine the height of the part at
each end, and subtract the small end from the large end. Then, divide the difference by the length
of the part in inches and multiply by 12 to find the taper per foot. In Figure 1, the part is 16 in.
long, with the larger diameter measuring 4 in. and the smaller diameter measuring 2 inches.
Subtract 2 from 4 to find a difference of 2. Divide 2 by the length of the part, 16, (2 / 16 = 0.125)
to find that the amount of taper is 0.125 for each inch of the part. Finally, multiply 0.125 by 12
(0.125 x 12 = 1.500) to find that 1.500 is the part’s taper per foot.
There are two ways to see a taper on a shop print. One specification may use the TPF, and the
other may require you to find the included angle of the taper, as shown in Figure 2. Trig ratios
are often used to find the included angle of the taper. The accuracy of the taper or the angle of the taper will affect the quality of the mating part that a machinist manufactures.
Figure 1. The taper per foot (TPF) is the
difference in the diameter of a part along 1 foot
of the part ’s length.
Figure 2. A shop print may require you to find
the included angle of the taper.
Lesson: 12/17
Taper per Foot: Sample Problem
For the cylindrical tapered part in Figure 1, the print lists 3.8904 for the length of the taper, as well
as 3.0000 and 1.3327 for the two diameters. To find the taper per foot, first find the difference
between the two known diameters (3.0000 – 1.3327 = 1.6673), as you can see in Figure 2. Next,
take this difference and divide it by 2 (1.6673 / 2 = 0.8336) to find the taper on one half or side of
the part. You can imagine a centerline drawn through the part to separate each side of the taper.
Remember, because this taper is symmetrical, whatever you do to one side of the taper you can
do to the other.
After you divide the diameter difference by 2, you can also draw in two imaginary lines to form two
right triangles, as you can see in Figure 3. You can place the value of 0.8336 on each leg of the two
triangles. From this point, you can work with one side of the taper.
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Figure 1. In this cylindrical tapered part, you
are given both diameters and the length of the
Lesson: 12/17
Taper per Foot: Sample Problem
For the cylindrical tapered part in Figure 1, the print lists 3.8904 for the length of the taper, as well
as 3.0000 and 1.3327 for the two diameters. To find the taper per foot, first find the difference
between the two known diameters (3.0000 – 1.3327 = 1.6673), as you can see in Figure 2. Next,
take this difference and divide it by 2 (1.6673 / 2 = 0.8336) to find the taper on one half or side of
the part. You can imagine a centerline drawn through the part to separate each side of the taper.
Remember, because this taper is symmetrical, whatever you do to one side of the taper you can
do to the other.
After you divide the diameter difference by 2, you can also draw in two imaginary lines to form two
right triangles, as you can see in Figure 3. You can place the value of 0.8336 on each leg of the two
triangles. From this point, you can work with one side of the taper.
Figure 1. In this cylindrical tapered part, you
are given both diameters and the length of the
taper.
Figure 2. To find the taper per foot, you must
first find the difference between the two
diameters.
Figure 3. You can draw two imaginary triangles
in the taper.
Lesson: 13/17
Taper per Foot: Solution
Consider the original print in Figure 1. You are working with one side of the taper to find the TPF.
You must divide the length of one leg on the right triangle by the length of the taper (0.8336 /
3.8904 = 0.2143), as shown in Figure 2. This gives you a ratio of 0.21428, which represents the
relationship of the sides of this particular right triangle.
If you were to use the angle of this right triangle as the reference angle, and think of it as taking
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the
opposite
sideTooling
over the
adjacent
side,Reserved.
or tangent, you would get the same result (0.2143). Now,
because you need to find the TPF, you must convert from inches to feet. Consider the equation in
Figure 3. Multiply this number by 12 (0.2143 x 12 = 2.5713) to get 2.5713. Remember, this is only
Lesson: 13/17
Taper per Foot: Solution
Consider the original print in Figure 1. You are working with one side of the taper to find the TPF.
You must divide the length of one leg on the right triangle by the length of the taper (0.8336 /
3.8904 = 0.2143), as shown in Figure 2. This gives you a ratio of 0.21428, which represents the
relationship of the sides of this particular right triangle.
If you were to use the angle of this right triangle as the reference angle, and think of it as taking
the opposite side over the adjacent side, or tangent, you would get the same result (0.2143). Now,
because you need to find the TPF, you must convert from inches to feet. Consider the equation in
Figure 3. Multiply this number by 12 (0.2143 x 12 = 2.5713) to get 2.5713. Remember, this is only
the TPF for one side of the taper. You must then multiply this number by 2 (2.5713 x 2 = 5.1426)
to get a total TPF of 5.1426.
Figure 1. The taper in the original print.
Figure 2. You can work with one side of the
taper to find the TPF.
Figure 3. You must convert the ratio to feet
and multiply by 2 to get the total TPF.
Lesson: 14/17
Finding a Taper Angle: Sample Problem
Some shop prints require you to find the included angle of the taper instead of the TPF. Consider
the same sample problem in Figure 1 discussed in the previous two lessons. If your print calls for
you to find the included angle of the taper shown in Figure 2, you still must first find the difference
between the two known diameters, which is 1.6673. Then, divide this difference by 2 to get 0.8336.
Finally, divide 0.8336 by the length of the part, or 3.8904 to get a ratio of 0.2143.
Remember, 0.2143 represents the tangent ratio of the right triangle, which is shown in Figure 3.
You must find the included angle of the taper in degrees. To do this, enter 0.2143, press the 2nd
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key, and press TAN. The inverse tangent gives you an angle of 12.0944 in decimal degrees.
Keep in mind that the 12.0944° angle represents the angle for only one of the right triangles on Lesson: 14/17
Finding a Taper Angle: Sample Problem
Some shop prints require you to find the included angle of the taper instead of the TPF. Consider
the same sample problem in Figure 1 discussed in the previous two lessons. If your print calls for
you to find the included angle of the taper shown in Figure 2, you still must first find the difference
between the two known diameters, which is 1.6673. Then, divide this difference by 2 to get 0.8336.
Finally, divide 0.8336 by the length of the part, or 3.8904 to get a ratio of 0.2143.
Remember, 0.2143 represents the tangent ratio of the right triangle, which is shown in Figure 3.
You must find the included angle of the taper in degrees. To do this, enter 0.2143, press the 2nd
key, and press TAN. The inverse tangent gives you an angle of 12.0944 in decimal degrees.
Keep in mind that the 12.0944° angle represents the angle for only one of the right triangles on one side of the taper. To find the included angle for the entire taper, you must multiply 12.0944 by
2 to get an included angle of 24.1888, or 24 degrees.
Figure 1. The taper in the original print.
Figure 2. The shop print may require you to
find the included angle of the taper.
Figure 3. You must take the inverse tangent of
your ratio and multiply it by 2 to get the total
included angle of the taper.
Lesson: 15/17
Finding a Taper Angle: Sample Problem #2
Consider the taper in the sample print in Figure 1. You must solve for X, which is the total included
angle of the taper. You are given 6.2000 for the length of the taper, and 4.5000 and 0.7000 for
the two diameters of the taper. What steps should you follow to find the included angle?
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U, LLC.
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The
first step
is to
find the
difference
of Reserved.
the diameters (4.5000 – 0.7000 = 3.8000), as shown in
Figure 2. Next, divide this difference by 2 (3.8000 / 2 = 1.9000). This allows you to draw two lines:
one from the top of the 0.7000 diameter straight through the taper and one from the bottom of
Lesson: 15/17
Finding a Taper Angle: Sample Problem #2
Consider the taper in the sample print in Figure 1. You must solve for X, which is the total included
angle of the taper. You are given 6.2000 for the length of the taper, and 4.5000 and 0.7000 for
the two diameters of the taper. What steps should you follow to find the included angle?
The first step is to find the difference of the diameters (4.5000 – 0.7000 = 3.8000), as shown in
Figure 2. Next, divide this difference by 2 (3.8000 / 2 = 1.9000). This allows you to draw two lines:
one from the top of the 0.7000 diameter straight through the taper and one from the bottom of
the 0.7000 diameter through the taper, creating two right triangles, as Figure 3 shows. Remember,
because this is a cylindrical and symmetrical part, both sides of the taper are the same. Therefore,
the leg of both right triangles is 1.9000 in length. From this point, you can work with one side of
the taper.
Figure 1. In this print, you must find the total
included angle of the taper.
Figure 2. You must first find the difference
between the two known diameters.
Figure 3. You can now create two right
triangles in the taper.
Lesson: 16/17
Sample Problem #2: Solution
Consider the original taper in Figure 1. You found one leg of your right triangle. Now you must
solve for the included taper angle X. You know that each leg of the two right triangles equals
1.9000. First, you must find the angle A of one of the right triangles, which is shown in Figure 2. If
you consider this angle as your reference angle, you know that the opposite side is 1.9000, and
the adjacent side is the length of the part, or 6.2000. You know also that opposite over adjacent is
the tangent ratio. Therefore, you must divide (1.9000 / 6.2000 = 0.3065), as shown in Figure 3.
Next, because you must find the value for the angle in decimal degrees, press the 2nd key and the
TAN key. The inverse of the tangent will give you an angle of 17.0378 in decimal degrees. However,
remember that you need to find the included angle of the taper. Because both sides of the taper
are the same, take 17.0378 and multiply it by 2 (17.0378 x 2 = 34.0756) to get a total included
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angle
of 34
degrees.
Figure 1. The taper in the original print.
Lesson: 16/17
Sample Problem #2: Solution
Consider the original taper in Figure 1. You found one leg of your right triangle. Now you must
solve for the included taper angle X. You know that each leg of the two right triangles equals
1.9000. First, you must find the angle A of one of the right triangles, which is shown in Figure 2. If
you consider this angle as your reference angle, you know that the opposite side is 1.9000, and
the adjacent side is the length of the part, or 6.2000. You know also that opposite over adjacent is
the tangent ratio. Therefore, you must divide (1.9000 / 6.2000 = 0.3065), as shown in Figure 3.
Next, because you must find the value for the angle in decimal degrees, press the 2nd key and the
TAN key. The inverse of the tangent will give you an angle of 17.0378 in decimal degrees. However,
remember that you need to find the included angle of the taper. Because both sides of the taper
are the same, take 17.0378 and multiply it by 2 (17.0378 x 2 = 34.0756) to get a total included
angle of 34 degrees.
Figure 1. The taper in the original print.
Figure 2. You must find angle A on one side of
the taper.
Figure 3. You must find the tangent of your
ratio and divide it by 2 to find the total included
angle of the taper.
Lesson: 17/17
Summary
A right triangle is one of the most common shapes found in shop drawings. You can use the sides
of a right triangle to find information about its angles, and you can use the measurement of its
angles to find information about its sides. The angles and sides of a right triangle establish
trigonometric ratios. The most common ratios include sine, cosine, and tangent. The sine of an
angle is the opposite side divided by the hypotenuse. The cosine is the side opposite the angle
divided by the adjacent side. The tangent is the side opposite the angle divided by the adjacent
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side.
A common word to help you remember these relationships is SOHCAHTOA. Each letter in the word
Lesson: 17/17
Summary
A right triangle is one of the most common shapes found in shop drawings. You can use the sides
of a right triangle to find information about its angles, and you can use the measurement of its
angles to find information about its sides. The angles and sides of a right triangle establish
trigonometric ratios. The most common ratios include sine, cosine, and tangent. The sine of an
angle is the opposite side divided by the hypotenuse. The cosine is the side opposite the angle
divided by the adjacent side. The tangent is the side opposite the angle divided by the adjacent
side.
A common word to help you remember these relationships is SOHCAHTOA. Each letter in the word
represents the ratios. You can use these ratios to find a missing dimension or angle in a right
triangle.
A taper is commonly found in shop prints. A taper is a gradual decrease in the diameter of the part
Figure 1. Trig ratios allow you to find missing
from one end to another. On a print, a taper forms a right triangle, with the angle of the taper as angles and dimensions for right triangles.
your reference angle and the tapered edge as your hypotenuse. There are two ways to see a taper
on a shop print. You may need to find the taper per foot (TPF) or the included angle of the taper.
Trig ratios are often used to find the included angle of the taper.
Figure 2. SOHCAHTOA is a simple method that
allows you to visualize the relationship
between trig ratios.
Class Vocabulary
Term
Definition
Adjacent Side
CAH
Cosecant
Cosine
Cotangent
Hypotenuse
Included Angle
Minute
Opposite Side
The side next to the reference angle in a right triangle. The adjacent side cannot be the hypotenuse.
Cosine is adjacent over hypotenuse.
In a right triangle, the ratio of the length of the hypotenuse divided by the opposite side of the angle. Cosecant is
the reverse of sine.
In a right triangle, the ratio of the length of the side adjacent to the angle divided by the hypotenuse.
In a right triangle, the ratio of the length of the adjacent side divided by the length of the opposite side of the angle.
Cotangent is the reverse of tangent.
The longest side of a right triangle. The hypotenuse is always opposite the 90° angle in a right triangle.
The entire angle that contains the taper. Each edge of the taper forms a leg of the angle.
A unit of measurement used to describe angles. There are 60 minutes in 1 degree.
The side across from the reference angle in a right triangle.
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Reference Angle The measured, or known angle in a right triangle other than the 90° angle.
Right Triangle
A triangle containing one angle that measures exactly 90 degrees.
Class Vocabulary
Term
Definition
Adjacent Side
CAH
Cosecant
Cosine
Cotangent
Hypotenuse
Included Angle
Minute
Opposite Side
Reference Angle
Right Triangle
Secant
Second
The side next to the reference angle in a right triangle. The adjacent side cannot be the hypotenuse.
Cosine is adjacent over hypotenuse.
In a right triangle, the ratio of the length of the hypotenuse divided by the opposite side of the angle. Cosecant is
the reverse of sine.
In a right triangle, the ratio of the length of the side adjacent to the angle divided by the hypotenuse.
In a right triangle, the ratio of the length of the adjacent side divided by the length of the opposite side of the angle.
Cotangent is the reverse of tangent.
The longest side of a right triangle. The hypotenuse is always opposite the 90° angle in a right triangle.
The entire angle that contains the taper. Each edge of the taper forms a leg of the angle.
A unit of measurement used to describe angles. There are 60 minutes in 1 degree.
The side across from the reference angle in a right triangle.
The measured, or known angle in a right triangle other than the 90° angle.
A triangle containing one angle that measures exactly 90 degrees.
In a right triangle, the ratio of the length of the hypotenuse divided by the adjacent side of the angle. Secant is the
reverse of cosine.
A unit of measurement used to describe angles. There are 60 seconds in 1 minute and 60 minutes in 1 degree.
Sine
In a right triangle, the ratio of the length of the side opposite the angle divided by the hypotenuse.
SOH
Sine is opposite over hypotenuse.
SOHCAHTOA
Symmetrical
Tangent
Taper
Taper Per Foot
TOA
Trigonometric Ratio
A common phrase that helps visualize the relationship between the trigonometric ratios. Each letter represents the
ratios in order.
A quality in which all the features on either side of a point, line, or plane are identical. Both sides of a symmetrical
part have the same dimensions.
In a right triangle, the ratio of the length of the side opposite the angle divided by the length of the adjacent side.
A conical object with a gradual decrease in diameter from one end to another. On a shop print, a taper forms a right
triangle.
A measurement unit for a taper indicating the change in diameter for each foot along the taper's length.
Tangent is opposite over adjacent.
A ratio that describes a relationship between the sides and angles of a triangle.
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