# 4.3 Right Triangle Trigonometry

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
Right Triangle Trigonometry
Side
opposite
θ
hypotenuse
θ
Using the lengths of these 3 sides, we form six
ratios that define the six trigonometric
functions of the acute angle θ.
sine
cosine
tangent
cosecant
secant
cotangent
*notice each pair has a “co”
Trigonometric Functions
• Let θ be an acute angle of a right triangle.
sin  
cos  
tan  
hypotenuse
RECIPROCALS
hyp
csc  
opp
hyp
sec  
cot  
opp
opposite
θ
Evaluating Trig Functions
•
Use the triangle to find the exact values of the six trig functions of θ.
sin 𝜃 =
csc 𝜃 =
cos 𝜃 =
sec 𝜃 =
tan 𝜃 =
cot 𝜃 =
5
θ
4
3
•
sin 𝛽 =
csc 𝛽 =
cos 𝛽 =
sec 𝛽 =
tan 𝛽 =
cot 𝛽 =
Repeat for β. What do you notice?
Solving Right Triangles with Trig
•
Use the trig functions to find the
missing sides for θ = 57&deg;
5
h θ
θ
2.
Use the trig functions to find the
missing sides for θ = 32&deg;
1.
h
1.
•
a
17
θ
2.
2
x
θ
3.
θ
27
3.
x
o
θ
x
3
100
Special Right Triangles
45-45-90
30-60-90
45&deg;
60&deg;
2
1
2
1
45&deg;
1
30&deg;
3
Evaluating Trig Functions for 45&deg;
• Find the exact value of the following:
• sin 45&deg; =
• cos 45&deg; =
45&deg;
• tan 45&deg; =
45&deg;
Evaluating Trig Functions for 30&deg; and 60&deg;
• Find the exact values of the following:
• sin 60&deg; =
• sin 30&deg; =
•
cos 60&deg; =
• cos 30&deg; =
• tan 60&deg; =
• tan 30&deg; =
30&deg;
60&deg;
Sine, Cosine, and Tangent of Special Angles
Θ in Degrees
30&deg;
π
6
60&deg;
π
3
45&deg;
π
4
sin Θ
3
2
1
2
2
2
cos Θ
1
2
3
2
2
2
tan Θ
3
3
3
1