Trigonometric Identities & Examples

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Trig Functions Illustration

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Fundamental Trigonometric Identities

Reciprocal Identities

csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)

Quotient Identities

tan(θ) = sin(θ)/cos(θ)
cot(θ) = cos(θ)/sin(θ)

Pythagorean Identities

sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)

Sum & Difference Identities

sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)
cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)
tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b)

Negative Angle Identities

sin(-θ) = -sin(θ), cos(-θ) = cos(θ), tan(-θ) = -tan(θ)

Law of Sines

a/sin A = b/sin B = c/sin C

Law of Cosines

c² = a² + b² - 2ab cos C

Area of Oblique Triangle

Area = ½ ab sin C

Arc Length & Sector Area

s = rθ, A = ½ r²θ (θ in radians)

Tappable Example Problems

Example 1: Fundamental Identity Simplify: (1 - cos²(x)) / sin(x)
Solution: Use sin²(x) = 1 - cos²(x)
Result: sin(x)
Example 2: Verifying Identity Prove: (1 + tan²(x)) / sec²(x) = 1
Use identity: 1 + tan²(x) = sec²(x)
So: sec²(x)/sec²(x) = 1
Example 3: Cos(75°) Use sum identity: cos(45° + 30°)
cos(75°) = cos(45°)cos(30°) - sin(45°)sin(30°)
Answer: (\sqrt{6} - \sqrt{2}) / 4
Example 4: Sin(15°) Use difference identity: sin(45° - 30°)
= sin(45°)cos(30°) - cos(45°)sin(30°)
Answer: (\sqrt{6} - \sqrt{2}) / 4

Graphing Trigonometric Functions

General forms:

y = a sin(bx)
y = a cos(bx)
y = a tan(bx)
Amplitude: |a|
Period: 2π / b

Sine Graph

Right Triangle Basics

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

Right Triangle

Coordinate Plane Identities

On the unit circle:
sin(θ) = y, cos(θ) = x, tan(θ) = y/x