What you should learn
How to evaluate
trigonometric functions
of acute angles
Section 5.2 Right Triangle
Trigonometry
Objective: In this lesson you learned how
to evaluate trigonometric functions of acute
angles and how to use the fundamental
trigonometric identities.
I. The Six Trigonometric Functions
In the right triangle shown below, label the three
sides of the triangle relative to the angle
labeled θ as
(a) the hypotenuse,
(b) the opposite side, and
(c) the adjacent side.
Abbreviations of the six functions.
• Using the lengths of these three sides, six
ratios can be formed that define the
following six trigonometric functions of the
acute angle θ.
• List the abbreviation for each trigonometric
function.
Definitions of the six functions.
• Let θ be an acute angle of a right triangle. Define the six
trigonometric functions of the angle θ using opp = the length
of the side opposite θ, adj = the length of the side adjacent to
θ, and hyp = the length of the hypotenuse.
Definitions of the six functions.
The cosecant function is the reciprocal of the
_________ function. The cotangent function is the
reciprocal of the __________ function. The secant
function is the reciprocal of the ___________
function.
Side opposite θ
The Six Trigonometric Functions
θ
Side adjacent θ
opp
sin  
hyp
adj
cos  
hyp
hyp
csc  
opp
hyp
sec  
adj
opp
tan  
adj
adj
cot  
opp
Example 1:
• In the right triangle below, find sin θ, cos θ,
and tan θ.
Give the sines, cosines, and tangents of the
following special angles:
sin 
tan  
cos 
Cofunctions
Cofunctions of complementary angles are
equal
____________
. If θ is an acute angle, then:
cos 
sin 
cot 
tan 
csc 
sec 
II. Trigonometric Identities
Reciprocal Identities
What you should learn
How to use the
fundamental
trigonometric identities
List two quotient identities:
List three Pythagorean identities:
sin   cos   1
2
2
1  tan   sec 
2
2
1  cot   c sc 
2
2
III. Evaluating Trigonometric
Functions with a Calculator
• To evaluate the secant function with a
calculator, . . .
1
sec  
cos 
What you should learn
How to use a calculator
to evaluate trigonometric
functions
Example 2:
Use a calculator to evaluate
(a) tan 35.4°
(b) cos 3
Remember to change modes
(a) = .7106630094
(b) = -.9899924966
IV. Applications Involving Right
Triangles
• What does it mean to “solve a
right triangle?”
• An angle of elevation is . . .
• An angle of depression is . . .