1. 4.3 Right Triangle Trigonometry
Objectives:
Evaluate trigonometric functions of acute angles
Use trig identities
Evaluate trig functions with a calculator
Use trig functions to model and solve real life problems

2. Right Triangle Trigonometry
Using the lengths of these 3 sides, we form six ratios that define the six trigonometric functions of the acute angle θ.
sine cosecant
cosine secant
tangent cotangent
*notice each pair has a “co”
Side opposite θ
hypotenuse θ
Side adjacent to θ

3. Trigonometric Functions
Let θ be an acute angle of a right triangle.
hypotenuse
RECIPROCALS
opposite θ
adjacent

4. Evaluating Trig Functions
Use the triangle to find the exact values of the six trig functions of θ.
Repeat for β. What do you notice?
sin 𝜃= csc 𝜃=
cos 𝜃= sec 𝜃=
tan 𝜃= cot 𝜃=
sin 𝛽= csc 𝛽=
cos 𝛽= sec 𝛽=
tan 𝛽= cot 𝛽= 5 4 θ 3

5. Solving Right Triangles with Trig
Use the trig functions to find the missing sides for θ = 57°
Use the trig functions to find the missing sides for θ = 32° h 5 θ h 17 x θ 2 x θ θ a θ 27 o x
100 3 θ

6. Special Right Triangles
45°
60° 2 1 2 1
45°
30° 1 3

7. Evaluating Trig Functions for 45°
Find the exact value of the following:
sin 45° =
cos 45° =
tan 45° =
45°

8. Evaluating Trig Functions for 30° and 60°
Find the exact values of the following:
sin 60° = • sin 30° =
cos 60° = • cos 30° =
tan 60° = • tan 30° =
60°

9. Sine, Cosine, and Tangent of Special Angles
Θ in Degrees
30°
60°
45°
Θ in Radians
π 6
π 3
π 4
sin Θ
3 2
1 2
2 2
cos Θ
tan Θ
3 3 3 1

10. Applications of Right Triangle Trigonometry
Angle of elevation: the angle from the horizontal upward to the object
Angle of depression: the angle from the horizontal downward to the object

11. Word Problems
A surveyor is standing 50 feet from the base of a large tree. The surveyor measure the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

12. You are 200 yards from a river
You are 200 yards from a river. Rather than walk directly to the river, you walk 400 yards along a straight, diagonal path to the river’s edge. Find the acute angle θ between this path and the river’s edge.

13. Find the length c of the skateboard ramp.