Download 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
Side adjacent to 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.
opp
sin  
hyp
adj
cos  
hyp
opp
tan  
adj
RECIPROCALS
csc  
hyp
opp
hyp
sec  
adj
adj
cot  
opp
Evaluating Trig Functions
• Use the triangle to find the exact values of the
six trig functions of θ.
hypotenuse
4
θ
3
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 sin 45°, cos 45°, and
tan 45°
Evaluating Trig Functions for 30° and
60°
• Find the exact values of sin60°, cos 60°,
sin 30°, cos 30°
60°
30°
Sine, Cosine, and Tangent of Special
Angles

1
sin 30  sin 
6 2
0

2
sin 45  sin 
4
2
0

3
sin 60  sin 
3
2
0

3
cos 30  cos 
6
2
0

2
cos 45  cos 
4
2
0
cos 600  cos

3

1
2
tan 300  tan

tan 450  tan
tan 60  tan
0
6


4

3
1
3
1
 3
sin30° = ½ = cos60° (notice that 30° and 60°
are complementary angles)
sin(90° - θ) = cos θ
cos(90° - θ) = sin θ
tan(90° - θ) = cot θ
cot(90° - θ) = tan θ
sec(90° - θ) = csc θ
csc(90° - θ) = sec θ
Trig Identities
• Reciprocal Identities
1
sin  
csc 
csc  
1
sin 
1
cos  
sec 
sec  
1
cos 
1
tan  
cot 
cot  
1
tan 
Trig Identities (cont)
• Quotient Identities
sin 
tan  
cos 
• Pythagorean Identities
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc 2 
cot  
cos 
sin 
Applying Trig Identities
• Let θ be an acute angle such that sin θ = .6.
Find the values of (a) cos θ and (b) tan θ using
trig identities.
Using Trig Identities
• Use trig identities to transform one side of the
equation into the other (0 < θ < π/2)
a) cos θ sec θ = 1
b) (sec θ + tan θ)(secθ – tanθ) = 1
Evaluating Using the Calculator
• sin 63°
• tan (36°)
• sec (5°)
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 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.