5.1 notes: Fundamental Identities
Trig identities: are statements that are true for all values of the variable for which both sides of the
equation are defined. For example: tan θ = sin θ/cos θ or csc θ = 1/sin θ
Domain of validity: set of all true values for an identity
Basic Identities:
Reciprocal Identities:
csc θ =
sec θ =
Quotient identities:
tan θ =
cot θ =
sin θ =
cos θ =
cot θ =
Do p. 405 exploration 1
Pythagorean Identities:
cos2 θ + sin2θ = 1
1 + tan2 θ = sec2 θ
cot2 θ + 1 = csc2 θ
examples: evaluate without using a calculator. Use the Pythagorean identities
1. Find sec θ and csc θ if tan θ = 3 and cos θ >0
2. Find sin θ and tan θ if cos θ = .8 and tan θ < 0
Cofunction identities:
sin (
cos (
tan (
csc (
sec (
cot (
Odd-Even Identities:
sin (-x) = -sin x
csc (-x) = - csc x
cos (-x) = cos x
sec (-x) = sec x
examples: Use identities to find the value of the expression
3. If tan (
4. If cot (-θ) = 7.89, find tan (θ -
Examples: Use basic identities to simplify the expression:
5. cot x tan x
6. cot u sin u
7.
tan (-x) = - tan x
cot (-x) = - cot x
tan θ =
Examples: simplify the expressions to either 1 or -1 or a basic trig function
8. sec (-x) cos (-x)
9. cot (-x) tan (-x)
10.
11.
Examples: use the basic identities to change the expressions. Your answer will be a basic trig function
12. sin θ – tan θ cos θ + cos (
13.
Examples: combine the fractions and simplify to a multiple of a power of a basic trig function
14.
15.
Examples: write each expression in factored form as an algebraic expression of a single trig function
16. 1 – 2sin x + sin2 x
17. sin2 x +
examples: make the suggested trig substitution, and then use Pythagorean identities to write the resulting function as a
multiple of a basic trig function.
18.
19.