Name______________________________________________
Chapter 5 Analytic Trigonometry
Section 5.1 Using Fundamental Identities
Objective: In this lesson you learned how to use fundamental
trigonometric identities to evaluate trigonometric functions
and simplify trigonometric expressions.
I. Introduction (Page 374) What you should learn
How to recognize and
Name four ways in which the fundamental trigonometric write the fundamental
identities can be used: trigonometric identities
1) to evaluate trigonometric functions
2) to simplify trigonometric expressions
3) to develop additional trigonometric identities
4) to solve trigonometric equations
The Fundamental Trigonometric Identities
List six reciprocal identities: List six cofunction identities:
1) sin u = 1/(csc u) 1) sin(π/2 − u) = cos u
2) cos u = 1/(sec u) 2) cos(π/2 − u) = sin u
3) tan u = 1/(cot u) 3) tan(π/2 − u) = cot u
4) csc u = 1/(sin u) 4) cot(π/2 − u) = tan u
5) sec u = 1/(cos u) 5) sec(π/2 − u) = csc u
6) cot u = 1/(tan u) 6) csc(π/2 − u) = sec u
List two quotient identities: List six even/odd identities:
1) tan u = (sin u)/(cos u)
1) sin(− u) = − sin u
2) cot u = (cos u)/(sin u)
2) cos(− u) = cos u
List three Pythagorean identities:
3) tan(− u) = − tan u
2 2
1) sin u + cos u = 1
4) csc(− u) = − csc u
2 2
2) 1 + tan u = sec u
5) sec(− u) = sec u
2 2
3) 1 + cot u = csc u
6) cot(− u) = − cot u
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved. 99
100 Chapter 5 • Analytic Trigonometry
II. Using the Fundamental Identities (Pages 375−378)
What you should learn
Example 1: Explain how to use the fundamental trigonometric How to use the funda-
identities to find the value of tan u given that mental trigonometric
sec u = 2 . identities to evaluate
trigonometric functions,
simplify trigonometric
Use the Pythagorean identity 1 + tan2 u = sec2 u.
expressions, and rewrite
Substitute 2 for the value of sec u and solve for
trigonometric expressions
tan u.
Example 2: Explain how to use the fundamental trigonometric
identities to simplify sec x − tan x sin x .
Rewrite the expression in terms of sines and
cosines. Combine the resulting fractions to obtain
(1 − sin2 x)/(cos x). Using the Pythagorean identity
sin2 u + cos2 u = 1, replace the numerator with
cos2 x. Simplify the result to obtain cos x.
Additional notes
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
