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
Side adjacent to θ
*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  
adj
adj
cot  
opp
opposite
θ
adjacent
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°
5
h θ
θ
2.
Use the trig functions to find the
missing sides for θ = 32°
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°
60°
2
1
2
1
45°
1
30°
3
Evaluating Trig Functions for 45°
• Find the exact value of the following:
• sin 45° =
• cos 45° =
45°
• tan 45° =
45°
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° =
30°
60°
Sine, Cosine, and Tangent of Special Angles
Θ in Degrees
30°
π
6
60°
π
3
45°
π
4
sin Θ
3
2
1
2
2
2
cos Θ
1
2
3
2
2
2
tan Θ
3
3
3
1
Θ in Radians
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
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?
• 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.
• Find the length c of the skateboard ramp.