Download Do 7.1

Chapter 7
Section 7.1
Fundamental Identities
Trigonometric Relations
The six trigonometric functions are related in many different ways. Several of these are quite useful for solving different
problems, finding values for the trigonometric functions or solving trigonometric equations.
Fundamental Trigonometric Identities: (They form the building blocks for many other identities.)
Reciprocal Identities:
1
csc x 
sin x
1
sec x 
cos x
1
cot x 
tan x
sin x
tan x 
cos x
cos x
cot x 
sin x
Pythagorean Identities:
sin x  cos x  1 tan x  1  sec x 1  cot x  csc x
2
2
2
2
2
2
Even/Odd Identities:
sin(  x)   sin x cos(  x)  cos x tan(  x)   tan x
Co-function Identities:
sin 2  x   cos x tan 2  x   cot x sec2  x   csc x
cos2  x   sin x cot 2  x   tan x csc2  x   sec x
Simplifying Trigonometric Expressions
It is often useful to be able to simplify a trigonometric expression. For example, when you are trying to solve an
equation that involves trigonometric functions. There are many ways to do this but the two that are used very
often are:
1. Change to sines and cosines.
2. Combine fractions and expressions where possible.
Here are some examples. (Notice these are expressions! (i.e. there is no equal sign).)
Simplify
Simplify
tan x csc x
sin x 1  cot x
sin x 1
cos x sin x
1
cos x
sec x

2
2
2
2
sin x csc x
1
2
sin x
2
sin x
1

Simplify
csc x  sin x
cot x
1
sin x
 sin x
cos x
sin x
 sin1 x  sin x sin x
cos x
sin x
sin x
1  sin x
cos x
2
2
cos x
cos x
cos x
Find the value for sin 𝜃 if the sec 𝜃 =
and tan 𝜃 < 0.
7
2
First need to figure out the sign. Since
secant is positive 𝜃 is in either quadrants
I or IV, and tangent is negative 𝜃 is either
in II or IV. The angle 𝜃 must be in
quadrant IV where sine is negative.
1
2
cos 𝜃 =
=
sec 𝜃 7
sin2 𝜃 + cos2 𝜃 = 1
4
45
sin 𝜃 = 1 − cos 𝜃 = 1 −
=
49 49
2
2
45
45
sin 𝜃 = −
=−
49
7
Find the values of all trigonometric functions if
−8
cot 𝜃 = and sin 𝜃 > 0.
5
Sine positive in I or II and Cotangent negative
in II or IV so 𝜃 is in quadrant II.
Reciprocal
−5
tan 𝜃 =
8
89
sec 𝜃 = −
sin 𝜃 =
8
89
5
Pythagorean
csc 2 𝜃 = 1 + cot 2 𝜃
64 89
2
csc = 1 +
=
25 25
89
89
csc 𝜃 = ±
=
5
5
cos 2 𝜃 = 1 − sin2 𝜃
25 64
2
cos = 1 −
=
89 89
8
8
cos 𝜃 = ±
=−
89
89
Any one trigonometric function can be expressed as some sort of combination of another
trigonometric function. The Identities allow you to do this conversion.
Example
Write the csc 𝑥 in terms of the cos 𝑥. (This means find out what the csc 𝑥 is equal to but only
using cos 𝑥)
1
csc 𝑥 =
sin 𝑥
Using the Pythagorean Identity sin 𝑥 can be written in terms of cos 𝑥.
sin2 𝑥 = 1 − cos2 𝑥
sin 𝑥 = ± 1 − cos2 𝑥
Substitute back:
csc 𝑥 =
1
± 1 − cos2 𝑥