01 Key Concepts
Pythagorean Identity
sin^2(theta) + cos^2(theta) = 1, true for every angle theta, derived directly from the unit circle.
Reciprocal Identities
csc(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), cot(theta) = 1/tan(theta).
Quotient Identity
tan(theta) = sin(theta)/cos(theta).
Even-Odd Identities
cos(-theta) = cos(theta) (cosine is even). sin(-theta) = -sin(theta) (sine is odd).
Using Identities to Simplify
Substitute known identities into an expression to rewrite it in a simpler or more useful equivalent form.
02 Key Formulas
- sin^2(theta) + cos^2(theta) = 1
- tan(theta) = sin(theta)/cos(theta)
- csc(theta) = 1/sin(theta)
- sec(theta) = 1/cos(theta)
03 Solved Examples
Example 1 If sin(theta) = 0.6, find cos(theta) using the Pythagorean identity (assume theta is in Quadrant I).
- sin^2+cos^2=1, so cos^2 = 1 - sin^2 = 1 - 0.36 = 0.64.
- cos(theta) = sqrt(0.64).
Answer: cos(theta) = 0.8
Example 2 Simplify sin(theta)*csc(theta).
- Recall csc(theta) = 1/sin(theta).
- sin(theta) * [1/sin(theta)] = 1.
Answer: 1
Example 3 If cos(theta) = 0.5, find sec(theta).
- Recall sec(theta) = 1/cos(theta).
- 1/0.5.
Answer: sec(theta) = 2
04 Practice Questions
1If sin(theta)=0.8, find cos(theta) (Quadrant I).
0.6
2Simplify cos(theta)*sec(theta).
1
3If sin(theta)=0.5, find csc(theta).
2
4What is the Pythagorean identity?
sin^2(theta) + cos^2(theta) = 1
5Is cosine an even or odd function?
Even, since cos(-theta)=cos(theta)
📄 Trigonometric Identities — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.