Download Lesson 1 – Trigonometric Functions Any two right triangles, with one

UNIT 6 NOTES
GEOMETRY B
Lesson 1 – Trigonometric Functions
1. I CAN state the trig ratios of a right triangle
2. I CAN explain why any right triangle yields the same trig values
3. I CAN explain the relationship of sine and cosine with complementary angles
Any two right triangles, with one other angle congruent, are
similar by AA Similarity.
This means that their side lengths are
.
These ratios between the side lengths are trigonometric ratios.
In a right triangle,
The sine of an angle is the ratio
The cosine of an angle is the ratio
The tangent of an angle is the ratio
Ex. Write each trigonometric ratio as a fraction.
sinJ 
cos A 
cosJ 
tanB 
tanK 
sinB 
UNIT 6 NOTES
GEOMETRY B
UNIT 6 NOTES
GEOMETRY B
Lesson 2 – Finding Sides of Right Triangles
6. I CAN solve right triangles using trig functions
We learned yesterday that these are the trigonometric ratios.
We now know that the trigonometric ratios are not dependent on the sides, but the ratios.
Therefore, there is one fixed value for every angle, from 0 to 90.
Your scientific (or graphing) calculator knows the values of the sine, cosine and tangent of all of
these angles. Depending on your calculator, you should have [SIN], [COS], and [TAN] buttons.
Use these to find the sine, cosine, and tangent of any acute angle.
Find the indicated trigonometric value, using your calculator.
sin 78°
cos 20°
tan 15°
Caution!
Be sure your
calculator is
in degree
mode, not
radian mode.
One application of the trigonometric ratios is to use them to find the missing sides of a right
triangle. All you need is one angle, other than the right angle, and one side.
Find each missing length. Round to the same decimal place as the given side.
AB
BC
QR
QP
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DF
.
GEOMETRY B
DE
UNIT 6 NOTES
GEOMETRY B
Lesson 3 – Finding Angles of Right Triangles
7. I CAN solve right triangles using inverse trig functions
Recall that these are the trigonometric ratios.
What happens if you have the sides of the triangle, but don’t have the angles?
The inverse trig functions are used to find the angle, with a given ratio.
Another application of the trigonometric ratios is to use them to find the missing angles of a right
triangle. All you need is the right angle and two side lengths.
Find each missing angle. Round to the nearest tenth of a degree.
mA
mB
mC
mB
UNIT 6 NOTES
GEOMETRY B
Solving Right Triangles
When given
pieces of information, one of those pieces being a right angle, you can
solve ANY right triangle, using the trigonometric functions and/or the inverse trigonometric
functions.
Solve each triangle.
mA
BC
AB
ST
m R
m T
UNIT 6 NOTES
GEOMETRY B
Lesson 4 – Angles of Elevation & Depression
8. I can use angles of elevation to find desired measurements
9. I can use angles of depression to find desired measurements
An angle of elevation and angle of depression are formed with the
_
____.
An inquisitive math student is standing 25 feet from the base of the
Washington Monument. The angle of elevation from her horizontal line of
sight is 87.4°. If her “eye height” is 5ft, how tall is the monument?
The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the
shadow to the top of the Space Needle is 70º, how tall is the Space Needle? Round to the
nearest meter.
UNIT 6 NOTES
GEOMETRY B
Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the airport to the
plane is 29°. What is the horizontal distance between the plane and the airport? Round to the
nearest foot.
Suppose a forest ranger in a 90-foot observation tower sees a fire and the angle of depression
to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot.
A restaurant needs to build a wheelchair ramp for its customers. The angle of elevation for a
ramp is recommended to be 5°. If the vertical distance from the sidewalk to the front door is two
feet, what is the horizontal distance that the ramp will take up? How long will the ramp be?
Round your answers to the nearest hundredth.
UNIT 6 NOTES
GEOMETRY B
Lesson 5 – The Law of Sines
10. I can solve triangles using the Law of Sines
As we’ve discussed before, you may only use SOHCAHTOA on right triangles.
To find parts of non-right triangles, or
Cosines can be used.
To solve any triangle, you must know
triangles, the Law of Sines and Law of
parts of the triangle.
Use the Law of Sines to find the missing part. Round side lengths the same as the given sides
and round angles to the nearest tenth.
Find NP.
Find MN.
UNIT 6 NOTES
GEOMETRY B
Find AB.
Find AC.
Find mQ .
Find mX .
UNIT 6 NOTES
GEOMETRY B
Lesson 6– The Law of Cosines
11. I can solve triangles using the Law of Cosines
To solve any triangle, you must know
parts of the triangle.
Look at the two triangles. Why can’t Law of Sines be used find a missing part with the given
information?
These triangles require the use of the Law of Cosines to solve them.
Use the Law of Cosines to find the missing part. Round side lengths the same as the given
sides and round angles to the nearest tenth.
Find XZ.
UNIT 6 NOTES
Find BC.
Find mS .
Find mP .
GEOMETRY B
UNIT 6 NOTES
GEOMETRY B
Lesson 7 – Special Right Triangles
4. I CAN apply the 30-60-90 right triangle theorem
5. I CAN apply the 45-45-90 right triangle theorem
In this lesson, we are going to establish the relationships in a 45-45-90 triangle and a 30-60-90
triangle. These are very common relationships and will be used in the last lesson to establish
“special” trig values.
45-45-90 Triangle:
30-60-90 Triangle:
For each triangle, fill in the missing sides and angles. Leave your answers in EXACT RADICAL
FORM.
UNIT 6 NOTES
GEOMETRY B
UNIT 6 NOTES
GEOMETRY B
Lesson 8– The Unit Circle
13. I can state the cosine, sine ordered pairs on the unit circle
Using the special triangles from Lesson 7 and the right triangle definitions of the trigonometric
functions, find the following without your calculator.
sin30 =
sin 45 =
sin60 =
cos30 =
cos 45 =
cos 60 =
tan30 =
tan 45 =
tan 60 =
The unit circle is a circle with a radius of one unit, centered at the origin. We are only going to
focus on the first quadrant, in this class.