• identity
• trigonometric identity
• cofunction
• odd-even identities
BELLRINGER: Define each word in your notebook.
Use Reciprocal and Quotient Identities
A. If , find sec θ.
Divide.
Reciprocal Identity
Answer:
Use Reciprocal and Quotient Identities
B. If and , find sin x.
Reciprocal Identity
Quotient Identity
Substitute for cos x.
Use Reciprocal and Quotient Identities
Answer:
Divide.
Multiply each side by .
Simplify.
Use Pythagorean Identities
If cot θ = 2 and cos θ < 0, find sin θ and cos θ.
cot 2 θ + 1= csc
2 θ Pythagorean Identity(2)
2 + 1= csc 2 θ
cot θ = 25 = csc
2 θ Simplify.
Use the Pythagorean Identity that involves cot θ.
= csc θ Take the square root of each side. Reciprocal Identity
Solve for sin θ.
Use Pythagorean Identities
Since is positive and cos θ < 0, sin θ
must be negative. So . You can
then use this quotient identity again to find cos θ.
Quotient Identity
cot θ = 2 and
Multiply each side by .
Use Pythagorean Identities
So,
Check sin 2 θ + cos 2 θ= 1Pythagorean Identity
Answer:
Simplify.
Use Cofunction and Odd-Even Identities
Simplify.cos x = –0.75Cofunction Identity
If cos x = –0.75, find
Odd-Even Identity
Factor.
Use Cofunction and Odd-Even Identities
Answer: 0.75
So, = 0.75.
Simplify by Rewriting Using Only Sine and Cosine
Solve Algebraically
Simplify .
Simplify.
Multiply.
Pythagorean Identity
So, = cos x.
Simplify by Factoring
Simplify cos x tan x – sin x cos 2 x.
Solve Algebraicallycos x tan x – sin x cos
2 x Original expression
Factor.
Multiply.
Quotient Identity
So, cos x tan x – sin x cos 2 x = sin3 x.
Pythagorean Identity
Simplify.= sin3x
Simplify by Combining Fractions
Common denominator
Multiply.
Add the numerators.
Simplify.
Pythagorean Identity
Simplify .
Answer: – 2 sec2 x – 2 sec2 x
Simplify by Combining Fractions
Reciprocal Identity
Reciprocal Identity
–2csc2 x.
Divide out common factor.
Reciprocal and Quotient Identities
Rewrite to Eliminate Fractions
Rewrite as an expression that does not
involve a fraction.
Pythagorean Identity
Reciprocal Identity
Quotient Identity
Reciprocal Identity
Rewrite to Eliminate Fractions
Answer: tan 2 x
So, = tan2 x.