Grade: Grade 11 Subject: Mathematics Unit: Advanced Trigonometry SAT: Geometry+Trigonometry ACT: Math

Trigonometric Identities

Learn

What Are Trigonometric Identities?

A trigonometric identity is an equation involving trig functions that is true for all values where both sides are defined. Identities are used to simplify expressions, solve equations, and prove other mathematical statements.

Fundamental 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

The Big Three

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

The second and third come from dividing the first by cos²θ and sin²θ respectively.

Even-Odd Identities

Even functions (symmetric about y-axis):

cos(-θ) = cos θ

sec(-θ) = sec θ

Odd functions (symmetric about origin):

sin(-θ) = -sin θ

tan(-θ) = -tan θ

csc(-θ) = -csc θ

cot(-θ) = -cot θ

Sum and Difference Formulas

Sine Sum/Difference

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

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

Cosine Sum/Difference

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

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

Tangent Sum/Difference

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

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

Double Angle Formulas

sin(2θ) = 2 sin θ cos θ

cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

tan(2θ) = 2 tan θ / (1 - tan²θ)

Half Angle Formulas

sin(θ/2) = ±√[(1 - cos θ)/2]

cos(θ/2) = ±√[(1 + cos θ)/2]

tan(θ/2) = (1 - cos θ)/sin θ = sin θ/(1 + cos θ)

The ± depends on the quadrant of θ/2.

Proving Identities: Strategies

  1. Work with one side only - Transform one side to match the other
  2. Start with the more complex side - Usually easier to simplify
  3. Convert to sines and cosines - Often reveals simplifications
  4. Factor when possible - Look for common factors or patterns
  5. Use Pythagorean identities - Convert between sin²θ, cos²θ, and 1
  6. Multiply by conjugates - Useful for expressions like 1 + sin θ or 1 - cos θ

Examples

Example 1: Using Pythagorean Identity

Problem: Simplify: (1 - cos²θ)/sin θ

Solution:

Step 1: Use sin²θ + cos²θ = 1, so 1 - cos²θ = sin²θ

= sin²θ / sin θ

= sin θ

Example 2: Proving an Identity

Problem: Prove: tan θ + cot θ = sec θ csc θ

Solution: Start with the left side, convert to sin and cos.

tan θ + cot θ = sin θ/cos θ + cos θ/sin θ

Step 2: Find common denominator

= (sin²θ + cos²θ)/(cos θ sin θ)

Step 3: Use Pythagorean identity

= 1/(cos θ sin θ)

= (1/cos θ)(1/sin θ)

= sec θ csc θ ✓

Example 3: Sum Formula Application

Problem: Find the exact value of sin(75°).

Solution:

Step 1: Write 75° = 45° + 30°

Step 2: Apply sin(A + B) = sin A cos B + cos A sin B

sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°)

= (√2/2)(√3/2) + (√2/2)(1/2)

= √6/4 + √2/4

= (√6 + √2)/4

Example 4: Double Angle Formula

Problem: If sin θ = 3/5 and θ is in Quadrant I, find sin(2θ).

Solution:

Step 1: Find cos θ using sin²θ + cos²θ = 1

cos²θ = 1 - (3/5)² = 1 - 9/25 = 16/25

cos θ = 4/5 (positive since QI)

Step 2: Apply sin(2θ) = 2 sin θ cos θ

sin(2θ) = 2(3/5)(4/5) = 24/25

Example 5: Simplifying with Identities

Problem: Simplify: (sec θ - 1)(sec θ + 1)

Solution:

Step 1: Recognize difference of squares pattern

= sec²θ - 1

Step 2: Use Pythagorean identity 1 + tan²θ = sec²θ

sec²θ - 1 = tan²θ

Practice

1. Simplify: sin θ cot θ

2. Simplify: (1 + tan²θ) cos²θ

3. Prove: sin²θ - cos²θ = 2sin²θ - 1

4. Find the exact value of cos(105°) using sum/difference formulas.

5. If cos θ = -5/13 and θ is in Quadrant III, find sin(2θ).

6. Simplify: (1 - sin²θ)(1 + tan²θ)

7. Prove: (sin θ + cos θ)² = 1 + sin(2θ)

8. Find tan(2θ) if tan θ = 3/4.

9. Simplify: sin(π/2 - θ)

10. If cos θ = 3/5 and θ is in Quadrant IV, find cos(θ/2).

Click to reveal answers
  1. cos θ
  2. 1 (since sec²θ · cos²θ = 1)
  3. sin²θ - cos²θ = sin²θ - (1 - sin²θ) = 2sin²θ - 1 ✓
  4. (√2 - √6)/4
  5. 120/169
  6. 1 (since cos²θ · sec²θ = 1)
  7. sin²θ + 2sinθcosθ + cos²θ = 1 + 2sinθcosθ = 1 + sin(2θ) ✓
  8. 24/7
  9. cos θ (cofunction identity)
  10. √[(1 + 3/5)/2] = √(4/5)/√2 = 2/√5 = 2√5/5

Check Your Understanding

1. Why are there three forms of the double angle formula for cosine?

Show answer

The forms cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ are all equivalent (you can derive each from the others using sin²θ + cos²θ = 1). Different forms are useful in different situations - use whichever makes your calculation simplest based on what's given.

2. How do you remember the sum formula for cosine versus sine?

Show answer

A common mnemonic: Cosine means "cosines together, sines together" (cos A cos B ∓ sin A sin B), while sine means "mix them up" (sin A cos B ± cos A sin B). Also note that cos(A + B) has a minus sign (opposites), while sin(A + B) has a plus sign (same sign as the operation).

3. When proving an identity, why shouldn't you work with both sides simultaneously?

Show answer

Working with both sides assumes the identity is already true (which is what you're trying to prove). The correct method is to transform one side independently until it matches the other. Think of it as a one-way path, not meeting in the middle.

4. What makes the Pythagorean identity sin²θ + cos²θ = 1 so fundamental?

Show answer

It comes directly from the Pythagorean theorem applied to the unit circle: if (x, y) is on the unit circle, then x² + y² = 1, and since x = cos θ and y = sin θ, we get cos²θ + sin²θ = 1. This identity connects all trig functions and is the source of many other identities.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review