Introduction To Trigonometric Identities
Introduction To Trigonometric Identities
Introduction To Trigonometric Identities
INTRODUCTION TO
TRIGONOMETRIC IDENTITIES
Learning Objectives
1. determine whether a trigonometric
equation is an identity or a
conditional equation
2. derive the fundamental
trigonometric identities
If W(x) = (a,b), where W is the wrapping
function
Quotient Identities
sin x cos x
tan x = cot x =
cos x sin x
If W(x) = (a,b), where W is the wrapping
function
Reciprocal Identities
1 1
csc x = sec x =
sin x cos x
1
cot x =
tan x
If W(x) = (a,b), where W is the wrapping
function
Pythagorean Identities
2 2
cos x + sin x = 1
2 2
1 + tan x = sec x
2 2
1 + cot x = csc x
EXAMPLE:
24
If tan x = and sin x < 0,
7
find all the other circular function
values at x
SOLUTION:
𝟕
cot x =
𝟐𝟒
72+242 25
sec x = ± 1 + tan2 x =± = ±
72 7
𝟐𝟓 𝟕
W(x) in third quadant, so sec x=− , cos x = −
𝟕 𝟐𝟓
24 −7 𝟐𝟒
sin x = (tan x)(cos x) = =−
7 25 𝟐𝟓
𝟐𝟓
csc x = −
𝟐𝟒
Odd-Even Identities
sin(−x) = − sin x csc(−x) = −csc x
cos(−x) = cos x sec(−x) = sec x
tan(−x) = − tan x cot(−x) = −cot x
That is, the sin, csc, tan, & cot functions are
odd, while cos & sec functions are even.
EXAMPLE
Is the given equation an identity or a
conditional equation?
cos x csc x = cot x
Solution:
1 cos x
cos x csc x = cos x ∙ = = cot x
sin x sin x
This is an identity.
EXAMPLE
Is the given equation an identity or a
conditional equation?
cos x = 1 − sin2 x
Solution:
A counter example is x = π
cos π = −1 ≠ 1 = 1 − sin2 π
This is a conditional