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Section 4.3
Right Triangle
Trigonometry
1
Objective
By following instructions students will be able to:
2

Evaluate trigonometric functions of acute angles.

Use the fundamental trigonometric identities.

Use a calculator to evaluate trigonometric functions.

Use trigonometric functions to model and solve real life
problems.
Recall: Soh-Cah-Toa
3
The Six Trigonometric Functions
4
Let  be an acute angle of a right triangle. Then the six trigonometric functions
of the angle
are defined as follows.

opp
sin  
hyp
adj
cos 
hyp
hyp
csc  
opp
opp 
tan  
adj

hyp
sec  
adj
adj
cot  
opp
Example 1:
Evaluating Trigonometric Functions
Find the exact values of the six
trigonometric functions of as
shown below.

hyp






3
4
5
Example 2:
Evaluating Trigonometric Functions
Find the exact values of


sin 45, cos45 , and tan 45.

6
Example 3:
Evaluating Trigonometric Functions of 30 and 60
degrees.
Use the equilateral triangle shown to find the exact values of
,
,
and
.
cos60 sin 30


cos30

7
sin 60
8
U-Try#1
1) Sketch a right triangle corresponding to the trig function of the acute
angle theta. Use the Pythagorean Theorem to determine the length
of the third side. Find the other trig functions of theta.
a)
3
cos  
7
b)
cot   5
1
3
sin
30

tan 30 
2) Use the given function
to find:
2 and
3

b) cos30
a) csc 30
c) cot 60


d) cot 30




Sines, Cosines, and Tangents of
Special Angles
9
1
2

3
cos30  cos 
6 2

3
tan 30  tan 
6 3

2
sin 45  sin 
4
2

2
cos 45  cos 
4 2

tan 45  tan  1
4
sin 30  sin

6



3 cos60  cos   1 tan 60  tan   3
sin 60  sin  
3
3 2
3 2

Cofunctions of Complementary
Angles are =.
sin( 90   )  cos
cos(90   )  sin 
tan(90   )  cot 
cot(90   )  tan 

sec(90   )  csc 
csc(90   )  sec 




10
11
Fundamental Trigonometric
Identities
Reciprocal Identities
1
sin  
csc 
1
cos 
sec 
1
tan  
cot 
1
csc  
sin 

1
sec  
cos 

1
cot  
tan 


Fundamental Trigonometric
Identities
12
Quotient Identities
sin 
tan  
cos
cos 
cot  
sin 
Pythagorean Identities
sin 2   cos2  1 1 tan 2   sec 2  1 cot 2   csc 2 




13
Example 4: Applying Trigonometric Identities

sin   0.6
Let be an acute angle such that
. Find the values of
(a)
and (b)
using trigonometric identities.
tan
cos


Candy Crush
14
1. Look for clues and
patterns (4 or 5 candies
that are the same).
2. More than one way to
complete the level
3. One objective: clear the
jellies.
15
Candy Crush
No more lives?
•
•
•
•
•
Get frustrated.
Wait
Look online for
help.
Ask friends for
help.
attempt again.
Trigonometric Proofs
16
1. Look for clues and
patterns (trig identities).
2. More than one way to
complete the problem.
3. One objective:
LHS=RHS
17
Trigonometric Proofs
No more lives?
•
•
•
•
•
Get frustrated.
Wait
Look online for
help.
Ask friends for
help.
attempt again.

Example 5:
Using Trigonometric Identities
18
Use trigonometric identities to transform one side of the equation into
the other.
a)
cos sec 1
b)

(sec   tan  )(sec   tan  )  1
Example 6:
Using a Calculator
Use a calculator to evaluate sec(540'12'')
.

19
Example 7:
Solving a Right Triangle
A surveyor is standing 50 feet from the base of a large tree. The
surveyor measures the angle of elevation to the top of the tree as
71.5 degrees. How tall is the tree?
20
Example 8:
Solving a Right Triangle
A person is 200 yards from a river. Rather than walk directly to the
river, the person walks 400 yards along a straight path to the river’s
edge. Find the acute angle between this path and the river’s
edge.


21
Example 9:
Solving a Right Triangle
A skateboard ramp requires a rise of one foot for each three feet of
horizontal length. Find the lengths of sides b and c and find the
measure of .


22
Revisit Objective
Did we…
23

Evaluate trigonometric functions of acute angles?

Use the fundamental trigonometric identities?

Use a calculator to evaluate trigonometric functions?

Use trigonometric functions to model and solve real life
problems?
Homework

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Pg 310 #s 1-67 ALL