Chapter 7
Trigonometric Identities and
Equations
7.1 BASIC TRIGONOMETRIC
IDENTITIES
Reciprocal Identities
1
csc 𝜃
1
csc 𝜃 =
sin 𝜃
1
tan 𝜃 =
cot 𝜃
sin 𝜃 =
cos 𝜃 =
1
sec 𝜃
1
sec 𝜃 = cos 𝜃
1
cot 𝜃 = tan 𝜃
These identities are derived in this manner
𝒚
𝟏
𝟏
sin 𝜽 = 𝟏 and csc 𝜽 = 𝒚 which gives you sin 𝜽 = 𝐜𝐬𝐜 𝜽
Quotient Identities
𝑠𝑖𝑛 𝜃
cos 𝜃
𝑐𝑜𝑠 𝜃
sin 𝜃
= tan 𝜃
= cot 𝜃
If using a unit circle as reference, these identities were derived using
𝑠𝑖𝑛 𝜃
cos 𝜃
𝑦
= 𝑥 = tan 𝜃
Pythagorean Identities
sin²𝜃 + cos²𝜃 = 1
tan²𝜃 + 1 = sec²𝜃
1 + cot²𝜃 = csc²𝜃
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] =
𝑡𝑎𝑛𝑎 ± 𝑡𝑎𝑛𝑏
1 ± 𝑡𝑎𝑛𝑎𝑡𝑎𝑛𝑏
This identity is used as both the sum and
difference identity.