Trigonometric Identities

What are Trigonometric Identities?

Identity: Equation that is true for all values of the variable

Use: Simplify expressions, verify equations, solve trig equations

Key skill: Recognize patterns and choose appropriate identity

Reciprocal Identities

Three pairs of reciprocal functions:

csc(θ) = 1/sin(θ)

sec(θ) = 1/cos(θ)

cot(θ) = 1/tan(θ)

Also:

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

Example 1: Simplify Using Reciprocal

Simplify: sin(θ) · csc(θ)

Apply reciprocal:

sin(θ) · 1/sin(θ) = 1

Answer: 1

Example 2: Convert to Sine and Cosine

Simplify: sec(θ) · cos(θ)

Apply:

1/cos(θ) · cos(θ) = 1

Answer: 1

Quotient Identities

Tangent and cotangent in terms of sine and cosine:

tan(θ) = sin(θ)/cos(θ)

cot(θ) = cos(θ)/sin(θ)

Example: Verify Quotient Identity

Show: tan(θ) · cot(θ) = 1

Substitute:

tan(θ) · cot(θ) = sin(θ)/cos(θ) · cos(θ)/sin(θ)
                = sin(θ)·cos(θ) / (cos(θ)·sin(θ))
                = 1

Pythagorean Identities

Three fundamental identities from unit circle:

sin²(θ) + cos²(θ) = 1 (most important!)

1 + tan²(θ) = sec²(θ)

1 + cot²(θ) = csc²(θ)

Note: sin²(θ) means (sin(θ))²

Example 1: Find Cosine from Sine

If sin(θ) = 3/5 and θ in Quadrant I, find cos(θ)

Use identity:

sin²(θ) + cos²(θ) = 1
(3/5)² + cos²(θ) = 1
9/25 + cos²(θ) = 1
cos²(θ) = 16/25
cos(θ) = ±4/5

Quadrant I (both positive): cos(θ) = 4/5

Answer: 4/5

Example 2: Verify Identity

Verify: tan²(θ) + 1 = sec²(θ)

Start with left side:

tan²(θ) + 1 = (sin(θ)/cos(θ))² + 1
            = sin²(θ)/cos²(θ) + 1
            = sin²(θ)/cos²(θ) + cos²(θ)/cos²(θ)
            = (sin²(θ) + cos²(θ))/cos²(θ)
            = 1/cos²(θ)
            = (1/cos(θ))²
            = sec²(θ) ✓

Even-Odd Identities

Cosine is even: cos(-θ) = cos(θ)

Sine is odd: sin(-θ) = -sin(θ)

Tangent is odd: tan(-θ) = -tan(θ)

Example: Simplify

Simplify: sin(-30°)

Use odd property:

sin(-30°) = -sin(30°) = -1/2

Answer: -1/2

Cofunction Identities

Complementary angles (add to 90°):

sin(θ) = cos(90° - θ)

cos(θ) = sin(90° - θ)

tan(θ) = cot(90° - θ)

Example: Evaluate

Find: sin(60°) using cofunction

Apply:

sin(60°) = cos(90° - 60°)
         = cos(30°)
         = √3/2

Sum and Difference Identities

For sine:

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

For cosine:

• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

For tangent:

• tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))
• tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))

Example 1: Exact Value

Find exact value: sin(75°)

Write as sum:

sin(75°) = sin(45° + 30°)

Apply sum identity:

= sin(45°)cos(30°) + cos(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6/4) + (√2/4)
= (√6 + √2)/4

Answer: (√6 + √2)/4

Example 2: Simplify

Simplify: cos(x)cos(y) - sin(x)sin(y)

Recognize sum identity for cosine:

= cos(x + y)

Double-Angle Identities

For sine: sin(2θ) = 2sin(θ)cos(θ)

For cosine (three forms):

• cos(2θ) = cos²(θ) - sin²(θ)
• cos(2θ) = 2cos²(θ) - 1
• cos(2θ) = 1 - 2sin²(θ)

For tangent: tan(2θ) = 2tan(θ)/(1 - tan²(θ))

Example 1: Double Angle

If sin(θ) = 3/5 (Quadrant I), find sin(2θ)

Find cos(θ) first:

cos²(θ) = 1 - (3/5)² = 16/25
cos(θ) = 4/5

Apply double-angle:

sin(2θ) = 2sin(θ)cos(θ)
        = 2(3/5)(4/5)
        = 24/25

Answer: 24/25

Example 2: Using Different Form

If cos(θ) = 1/3, find cos(2θ)

Use second form:

cos(2θ) = 2cos²(θ) - 1
        = 2(1/3)² - 1
        = 2(1/9) - 1
        = 2/9 - 9/9
        = -7/9

Answer: -7/9

Half-Angle Identities

Derived from double-angle identities:

sin(θ/2) = ±√[(1 - cos(θ))/2]

cos(θ/2) = ±√[(1 + cos(θ))/2]

tan(θ/2) = ±√[(1 - cos(θ))/(1 + cos(θ))]

Sign depends on quadrant of θ/2

Example: Half-Angle

Find exact value: cos(15°)

Write as half-angle:

cos(15°) = cos(30°/2)

Apply (15° in Q1, so positive):

= √[(1 + cos(30°))/2]
= √[(1 + √3/2)/2]
= √[(2 + √3)/4]
= √(2 + √3)/2

Verifying Identities

Strategy:

1. Work with more complicated side
2. Convert to sin and cos
3. Use known identities
4. Factor or combine fractions
5. Simplify until both sides match

Example: Verify

Verify: (1 - sin²(θ))/cos(θ) = cos(θ)

Start with left side:

(1 - sin²(θ))/cos(θ) = cos²(θ)/cos(θ)  [Pythagorean identity]
                     = cos(θ) ✓

Example 2: More Complex

Verify: tan(θ) + cot(θ) = sec(θ)csc(θ)

Convert to sin and cos:

LHS = sin(θ)/cos(θ) + cos(θ)/sin(θ)
    = (sin²(θ) + cos²(θ))/(sin(θ)cos(θ))
    = 1/(sin(θ)cos(θ))
    = (1/sin(θ)) · (1/cos(θ))
    = csc(θ) · sec(θ)
    = RHS ✓

Power-Reducing Identities

Used to reduce powers:

sin²(θ) = (1 - cos(2θ))/2

cos²(θ) = (1 + cos(2θ))/2

tan²(θ) = (1 - cos(2θ))/(1 + cos(2θ))

Example: Rewrite Without Powers

Rewrite sin²(x) without exponent

Apply:

sin²(x) = (1 - cos(2x))/2

Real-World Applications

Engineering: Signal processing, oscillations

Physics: Wave interference, energy calculations

Navigation: GPS calculations

Music: Harmonics and sound waves

Computer graphics: Rotation transformations

Practice