Download Trigonometric Ratios

Trigonometric Ratios
The word trigonometry comes from the Greek roots tri meaning three; gon meaning angle; and meter
meaning measure. Put it all together and one gets three angle measure. What has three angles? A
triangle does; hence the name. Trigonometric ratios then are ratios (comparison using division) of the 3
sides of a right triangle (right triangles only). The three most common ratios are the tangent, the sine,
and the cosine ratio. These ratios and their inverses can be used to find the lengths of unknown sides,
and the measures of unknown angles. Below is an explanation of these ratios.
Opposite- The side
opposite the given angle.
5.7
The hypotenuse will always be the hypotenuse,
because it is always opposite the 900 angle and
not the given angle. As seen in the examples
The opposite and the adjacent label for a side
can change depending upon which of the two
remaining angles is given.
Adjacent – The side
next to the given angle
10
8.2
Hypotenuse – The side
opposite the right angle
350
Adjacent – The side next to
the given angle
550
5.7
10
Hypotenuse – The side
opposite the right angle
Notice in the first triangle that the opposite side
is the shortest side, and the adjacent side is the
8.2
bottom side. That is because the angle given is
the smallest angle in the triangle and the side
Opposite- The side
farthest away from it and not touching it is the
opposite the given angle.
short side. The side touching it (besides the
hypotenuse) is the bottom side. In the second triangle the other angle is given (550). Now the side that
was the opposite to the 350 angle is adjacent to the 550 angle, and the side that was the adjacent to the
350 angle is now the opposite to the 550 angle.
Now that we have that cleared up, here are the ratios.
The Tangent Ratio =
Opposite
Adjacent
Sine Ratio =
Opposite
Hypotenuse
Cosine Ratio =
Adjacent
Hypotenuse
About this time you’re probably wondering, “Great what’s the use of this?” We’ll show you.
Trigonometric ratios can be used to find unknown lengths of sides.
These two triangles below are similar, therefore all of their corresponding angles are congruent.
The second triangle is twice the size of the first, so all of the side lengths are twice the
lengths of the sides in the first triangle. Not only that, but the sides within each
triangle have comparable ratios to the other.
40
20
For example the ratio of the short leg to the
20
hypotenuse in the first triangle is 10/20
10
0
which equals 0.5, while the same ratio
300
30
in the larger triangle is 20/40 which
17.32
34.64
equals 0.5 also. This is called the sine ratio.
It would be called the sine of 300 because that is the given angle and we divided the side opposite to
the angle by the hypotenuse to get this ratio. Now what would we do if we didn’t know the length of
Trigonometric Ratios
the hypotenuse but did have the given angle and the length of the side opposite the angle? Fortunately
someone with more knowledge than you or me has taught the calculator how to give us the ratio from a
given angle. Grab a scientific calculator (it will have trigonometric functions built in) and find the sin
of 300. Sin is the abbreviation for sine. In some calculators you will have to type 30 and then push the
sin button, followed by = ; while on others you will need to press sin first, and then type in the 30
followed by =. In either case the answer should be 0.5 (if not get help). What this means is that for all
right triangles with a 300 angle in them, if you take the side opposite to that angle and divide it by the
hypotenuse you will get a ratio of 0.5. You could set this up as seen below.
Sine 300 =
Opposite 
Hypotenuse
0.5 =
10
 0.5 =
Hypotenuse
10
20
The length of the Hypotenuse would be 20. (10  0.5 = 20. Had you tried
multiplying the ratio by the length of the hypotenuse you would have gotten 5 for an answer which
would have made no sense as the hypotenuse is always the longest side.)
What if the angle was 350 instead of 300? Just follow the same procedure but type in 35 instead of 30.
The length of the hypotenuse in this case should be 17.43 (approximately).
For the Cosine Ratio use
Cosine 300 =
Adjacent
Hypotenuse
We know in this case (300 angle) the hypotenuse was 20, so…..
Cosine 300 = .866  .866 =
Adjacent
 .866 =
20
17.32
20
The length of the Adjacent side would be 17.32 (20 * 8.66 Had you divided
by .866 you would have gotten 23.1 which would have made no sense as the adjacent cannot be
longer than the hypotenuse).
For the Tangent Ratio use
Tangent 300 =
Opposite
Adjacent
We already know the lengths so try dividing the opposite (10) by the adjacent 17.32). You should get
.5774 (rounded off) for an answer. Clear the calculator and use it to find the Tan of 300. It should be
the same number (very close anyway as the 17.32 was a rounded number). Had you not known one of
the sides you could have used the ratio to find it.
There is a trick to knowing when to divide and when to multiply using the ratio. No matter what, the
answer should make sense. The hypotenuse is always going to be the longest side. If you get an
answer in which the opposite or the adjacent is longer then you either need to try again or use a
different mathematical operation.
The Trick: (It works for all the ratios)
Note: The more decimal
places you include in the
ratio, the more accurate
your answers will be.
Divide by ratio
10
Tangent 300  .5774 =
Adjacent
Opposite
17.32
Multiply by ratio
Tangent 300  .5774 =
Trigonometric Ratios
For each triangle below find the sine, cosine, and tangent ratio of the given angle a, then for the given
angle b. (You may need to use the Pythagorean Theorem to find a missing side.)
1.
2.
3.
b
a
b
9
10
10
10 2
a
a
b
24
12
Find the lengths of the unknown sides. Then determine the area of each.
4.
5.
6.
?
10
?
20
380
580
650
?
?
?
?
12
7. How long will the boards need to be in order to make the following roof design? Don’t worry about
the thickness of the board in this case.
14 ft total
14 ft total
?
2 foot overhang
400
400
?
8. Find the length of side c.
c
700
15
2 foot overhang
Trigonometric Ratios
Finding Unknown Angles
So far we have been finding lengths of unknown sides by using a calculator to find a ratio that relates
to a given angle. Now we’re going to use the calculator to find the measures of unknown angles. If you
look at your calculator you will see above the sin in tiny little letters sin-1; you’ll see above the other
functions, cos-1 and tan-1. These are called inverse trigonometric functions, or arcsine, arccosine,
and arctangent (an arc in a circle defines the measure of an angle). Below is an example of how to
use this function.
In the triangle to the right the opposite to a is 3, while the
adjacent to a is 4. The tangent ratio is opposite/adjacent
so the tangent ratio of a is equal to 0.75.
b
5
3
On your calculator, push shift or 2nd (depending upon the
calculator) and then press tan; type in the 0.75, and then
press =. You should get 38.87 (rounded off). That is the
measure in degrees of angle a.
a
4
If that doesn’t work try typing in 0.75 first and then pressing shift or 2nd, followed by tan, and then =.
If you still don’t get 38.87 then find help.
Try the same procedure using the sine of a. The sine ratio is the opposite/hypotenuse, so the ratio is
3/5 or 0.6. This time use sin-1 instead. You should get the same angle measure for a, as the angle
doesn’t change just because you used a different trigonometric ratio. You would also get the same
answer if you used the cosine ratio (4/5 or 0.8) with cos-1.
Try finding the measure of angle b using the same procedures. Because 1800 – (900 + 38.870) = 51.130
you should come out with 51.10 (rounded off) for an answer. If you don’t, try again or get help.
(Notice that the cosine ratio for angle b is the same as the sine ratio for angle a, and visa versa.)
So, given any two sides in a right triangle one can find all the angle measures using the trigonometric
ratios and a scientific calculator. (In the old days we had to use tables of trigonometric ratios. Aren’t
you lucky!)
Trigonometric Ratios
Find the measures of the unknown angles (a and b) for each triangle.
1.
2.
b
a
13
10
7
b
a
12
3.
4.
a
13
a
28.28
15
b
20
b
5. Find the measure of angle a.
12
15
11
a
400
7