### 7.1 - 7.3 -- Trigonometric Identities and Equations

```Chapter 7
Trigonometric Identities and
Equations
7.1 BASIC TRIGONOMETRIC
IDENTITIES
Reciprocal Identities
1
csc
1
csc  =
sin
1
tan  =
cot
sin  =
cos  =
1
sec
1
sec  = cos
1
cot  = tan
These identities are derived in this manner

sin  =  and csc  =  which gives you sin  =
Quotient Identities

cos

sin
= tan
= cot
If using a unit circle as reference, these identities were derived using

cos

=  = tan
Pythagorean Identities
sin² + cos² = 1
tan² + 1 = sec²
1 + cot² = csc²
Opposite Angle Identities
sin [-A] = -sin A
cos [-A] = cos A
7.2 VERIFYING TRIGONOMETRIC
IDENTITIES
Tips For Verifying Trig Identities
• Simplify the complicated side of the equation
• Use your basic trig identities to substitute
parts of the equation
• Factor/Multiply to simplify expressions
• Try multiplying expressions by another
expression equal to 1
• REMEMBER to express all trig functions in
terms of SINE AND COSINE
7.3 SUM AND DIFFERENCE
IDENTITIES
Difference Identity for Cosine
Cos (a – b) = cosacosb + sinasinb
• As illustrated by the textbook, the difference
identity is derived by using the Law of Cosines
and the distance formula
Sum Identity for Cosine
Cos (a + b) = cos (a - (-b))
The sum identity is found by replacing -b with b
*Note*
If a and b represent the measures of 2
angles then the following identities apply:
cos (a ± b) = cosacosb ± sinasinb
Sum/Difference Identity For Sine
sinacosb + cosasinb = sin(a + b) – sum identity
for sine
If you replace b with (-b) you can get the
difference identity of sine.
sin (a – b) = sinacosb - cosasinb
Sum & Difference
Tan[a ± b] =
±
1 ±
This identity is used as both the sum and
difference identity.
```