5.1 Fundamental Identities
Name: ___________________
Objectives: Students will be able to use fundamental identities to simplify
trigonometric expressions and solve trigonometric equations.
Reciprocal Identities:
cscθ = 1
sinθ = 1
sinθ
cscθ
cosθ =
1
secθ =
cosθ
secθ
tanθ
1
1
cotθ =
1
tanθ
cotθ
Quotient Identities
tanθ = sinθ
cotθ = cosθ
cosθ
sinθ
Nov 25­3:23 PM
Pythagorean Identities:
sin2θ + cos2θ = 1
The other two Pythagorean Identities can be easily derived from
the one above.
-Divide the above equation by cos2θ:
-Divide the above equation by sin2θ:
Nov 25­3:42 PM
1
Cofunction Identities
sin(π/2 - θ) = cosθ
cos(π/2 - θ) = sinθ
tan(π/2 - θ) = cotθ
cot(π/2 - θ) = tanθ
sec(π/2 - θ) = cscθ
csc(π/2 - θ) = secθ
Odd-Even Identities
sin(-θ) = -sinθ
csc(-θ) = -cscθ
cos(-θ) = cos(θ)
sec(-θ) = sec(θ)
tan(-θ) = -tanθ
cot(-θ) = -cotθ
Nov 25­3:44 PM
Example Evaluate without a calculator. Use the Pythagorean
identities rather than reference triangles.
Find sinθ and tanθ if cosθ = 0.8 and tanθ < 0.
Example Use identities to find the value of the expression.
If cot(-θ) = 7.89, find tan(θ - π/2).
Nov 25­3:54 PM
2
Examples Use basic identities to simplify the expressions.
1.) sec(-x)cos(-x)
2.) cotβsinβ
3.) 1 - cos2θ
sinθ
Nov 25­3:58 PM
4.) sin3x + sinxcos2x
5.) (secx - 1)(secx + 1)
sin2x
Nov 25­4:05 PM
3
6.)
cosx -
1 - sinx
sinx
cosx
Nov 25­4:06 PM
7.) Find all values in [0,2π) that solve cos3x = cotx.
sinx
Nov 25­4:06 PM
4
8.) Find all the trigonometric solutions to 2sin2x + sinx = 1
Nov 25­4:11 PM
Assignment: Pages 451-452: #1-73 odd
Nov 25­4:12 PM
5