Right Triangle Trigonometry

```4.3 Right Triangle Trigonometry
Objectives:
Evaluate trigonometric functions of acute angles
Use the fundamental trigonometric identities
Use trigonometric functions to model and solve
real-life problems
Trigonometric Functions
Given a right triangle with acute angle 


Precalculus
opp
sin  
hyp
hyp
csc  
opp
cos  
hyp
hyp
sec  
opp
tan  


cot  
opp
4.3 Right Triangle Trigonometry
opp
hyp

2
Example 1
&middot; Evaluate the six trig
functions of the
angle  shown in the
right triangle
&middot; sin  =
5 
&middot; csc  =
13
&middot; cos  =
&middot; tan  =
&middot; sec  =
12
&middot; cot  =
Precalculus
4.3 Right Triangle Trigonometry
3
Example 2
&middot; Find the value of x for the right triangle
shown
30&deg;
15
x
Precalculus
4.3 Right Triangle Trigonometry
4
Example 3
&middot; Solve ∆ABC
c
A
Precalculus
b
B
62&deg;
6
C
4.3 Right Triangle Trigonometry
5
You Try
&middot; Solve ∆ABC
B
c
A

Precalculus
30&deg;
a
15
C
4.3 Right Triangle Trigonometry
6
Special Triangles
30&ordm;-60&deg;-90&deg;
60&deg;
1
Recall the conversions
2

60&deg;= 3

30&deg;= 6
30&deg;
3

1
sin 30  sin 
6 2



3
3 2

3

 1
cos30  cos 
cos60  cos 
2
6
3 2
 



1
3
3
tan 30  tan 
tan60  tan 

 3
6
3 1
3
3
Precalculus

4.3 Right
Triangle Trigonometry 
sin 60  sin

7
Special Triangles
45&ordm;-45&deg;-90&deg;
Recall the conversion
45&deg;

45&deg;= 4
2
1
45&deg;

1

sin 45  sin

4

1
2

2
2

Precalculus
1
2
cos45  cos 

4
2
2


 1
tan 45  tan   1
4 1
4.3 Right Triangle Trigonometry

8
Co-functions of
Complementary Angles
&middot; Look back at the sines and cosines for
the 30-60-90 triangle
&middot; Note that sin 30&deg;=cos 60&deg;
&middot; Co-functions of complementary angles
are equivalent
Precalculus
sin 90    cos
cos90    sin 
tan90    cot 
cot 90    tan 
sec90    csc 
csc90    sec 
4.3 Right Triangle Trigonometry
9
Trigonometric Identities
&middot; Reciprocal Identities



Precalculus
1
sin  
csc 
1
csc  
sin 
1
cos 
sec 
1
sec  
cos 
1
tan  
cot 


1
cot  
tan 
4.3 Right Triangle Trigonometry

10
Trigonometric Identities
&middot; Quotient Identities
sin 
tan  
cos

Precalculus
cos 
cot  
sin 

4.3 Right Triangle Trigonometry
11
Trigonometric Identities
&middot; Pythagorean Identities
sin 2   cos 2   1
1 tan 2   sec 2 



Precalculus
1 cot 2   csc 2 
Note: sin2 is (sin )2 not sin (2)
4.3 Right Triangle Trigonometry
12
Example 4
&middot; Given sin=0.6, find the value of cos
&middot; Given sin=0.6, find the value of tan
Precalculus
4.3 Right Triangle Trigonometry
13
Example 5
Use trigonometric identities to transform
one side of the equation into the other.
&middot; cossec=1
&middot; (sec+tan)(sec-tan)=1
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4.3 Right Triangle Trigonometry
14
Surveying
&middot; A surveyor is standing 50 ft from the
base of a large tree. The surveyor
measures the angle of elevation to the top
of the tree as 71.5&deg;. How tall is the tree?
Precalculus
4.3 Right Triangle Trigonometry
15
River Walk
&middot; You are 200 yards
from a river. Rather
than walking directly
to the river, you walk
400 yards along a
straight path to the
river’s edge. Find
the acute angle ө
between this path
and the river’s edge.
Precalculus
4.3 Right Triangle Trigonometry
16
Skateboarding
&middot; Find the length c of the skateboard ramp.
Precalculus
4.3 Right Triangle Trigonometry
17
Homework
&middot; Star Wars Worksheet
&middot; Page 285
44-54 even, 79-82 all
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4.3 Right Triangle Trigonometry
18
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