Aim for FivePrecalculus

Unit 3 · Topic 3.12

3.12 Equivalent Representations of Trigonometric Functions

Trig identities are equations that are true for every allowed angle. The Pythagorean identity comes from the unit circle, and the sum formulas for sine and cosine give the difference and double-angle formulas. You'll use identities to find exact values, rewrite expressions and solve equations.

Key terms

  • Pythagorean identity
  • sum identity
  • difference identity
  • double-angle identity
  • trig identity

Pythagorean identities

The point (cos θ, sin θ) is on the unit circle, x² + y² = 1. So sin² θ + cos² θ = 1 for every θ. (sin² θ means (sin θ)².)

Divide every term by cos² θ to get tan² θ + 1 = sec² θ. Divide by sin² θ to get 1 + cot² θ = csc² θ.

These let you find one trig value from another. If you know sin θ and the quadrant, cos θ = ±√(1 − sin² θ), with the sign from the quadrant.

They also link inverse functions: for 0 ≤ x ≤ 1, arcsin x = arccos(√(1 − x²)), because both name the same first-quadrant angle.

Sum and difference identities

sin(α + β) = sin α cos β + cos α sin β

cos(α + β) = cos α cos β − sin α sin β

Replacing β with −β, and using sin(−β) = −sin β and cos(−β) = cos β, gives the difference identities: sin(α − β) = sin α cos β − cos α sin β and cos(α − β) = cos α cos β + sin α sin β.

Double-angle identities

Setting β = α in the sum identities:

  • sin(2θ) = 2 sin θ cos θ
  • cos(2θ) = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ

Verifying identities and using them

To verify an identity, start with the more complicated side and transform it into the other, one justified step at a time. Useful moves: rewrite everything in sine and cosine, use a Pythagorean identity, factor, or combine fractions. Don't work on both sides at once as if it were an equation you already know is true.

Rewriting can make an equation solvable. sin(2θ) = sin θ mixes two different angles, but writing sin(2θ) = 2 sin θ cos θ lets you factor.

You won't be tested on half-angle identities, tangent sum and difference identities, or other identities not listed here. The Pythagorean identities and the sum, difference and double-angle identities for sine and cosine are fair game.

Example: to show (1 − cos² θ)/sin θ = sin θ wherever sin θ ≠ 0, start with the left side. By the Pythagorean identity, 1 − cos² θ = sin² θ, so the left side is sin² θ / sin θ = sin θ, which is the right side.

Worked examples

Try each one yourself first, then open the solution.

  1. Example 1

    An exact value from a difference

    Find the exact value of sin(π/12).

    Show the solution
    1. Step 1: π/12 = π/3 − π/4, and both of those have known values.
    2. Step 2: sin(π/3 − π/4) = sin(π/3)cos(π/4) − cos(π/3)sin(π/4).
    3. Step 3: = (√3/2)(√2/2) − (1/2)(√2/2) = √6/4 − √2/4.
    4. Step 4: Check the size: π/12 is 15°, a small angle, and (√6 − √2)/4 ≈ 0.259 is small and positive.

    Answer: sin(π/12) = (√6 − √2)/4.

  2. Example 2

    Double angles from one value

    sin θ = 3/5 and π/2 < θ < π. Find cos θ, sin(2θ) and cos(2θ).

    Show the solution
    1. Step 1: Pythagorean identity: cos² θ = 1 − 9/25 = 16/25, so cos θ = ±4/5. In quadrant II cosine is negative: cos θ = −4/5.
    2. Step 2: sin(2θ) = 2 sin θ cos θ = 2(3/5)(−4/5) = −24/25.
    3. Step 3: cos(2θ) = 1 − 2 sin² θ = 1 − 2(9/25) = 7/25.
    4. Step 4: Check: (−24/25)² + (7/25)² = (576 + 49)/625 = 1.

    Answer: cos θ = −4/5, sin(2θ) = −24/25, cos(2θ) = 7/25.

  3. Example 3

    Trap: dividing away solutions

    Solve sin(2θ) = sin θ for 0 ≤ θ < 2π.

    Show the solution
    1. Step 1: Rewrite: 2 sin θ cos θ = sin θ.
    2. Step 2: The trap is dividing both sides by sin θ, which gives cos θ = 1/2 and loses every solution where sin θ = 0.
    3. Step 3: Instead, move everything to one side and factor: 2 sin θ cos θ − sin θ = 0, so sin θ (2 cos θ − 1) = 0.
    4. Step 4: sin θ = 0 gives θ = 0 or π. cos θ = 1/2 gives θ = π/3 or 5π/3.

    Answer: θ = 0, π/3, π, 5π/3.

Common mistakes

  • Writing sin(α + β) = sin α + sin β. Use the sum identity.
  • Choosing the wrong sign for a square root from the Pythagorean identity. Let the quadrant decide.
  • Writing sin(2θ) = 2 sin θ. The double-angle identity is 2 sin θ cos θ.
  • Dividing by a trig expression that could be zero when solving.

On the exam

  • The symbolic-manipulation free-response question often asks you to rewrite a trig expression using these identities, then solve an equation on an interval. Name each identity as you use it.
  • Multiple-choice questions may give sin θ and a quadrant and ask for cos(2θ) or sin(2θ).

Connected topics

Videos

  • AP Precalculus – 3.12A Equivalent Representations of Trigonometric Functions

    The AlgebrosWatch on YouTube (opens in a new tab)

  • AP Precalculus – 3.12B Equivalent Representations of Trigonometric Functions

    The AlgebrosWatch on YouTube (opens in a new tab)

  • Trigonometric Identities in Under 3 mins (AP Precalculus Unit 3 Topic 3.12)

    Maximum InsightWatch on YouTube (opens in a new tab)

  • The Trigonometric Identities - AP Precalculus Topic 3.12 Part 1

    Michael Porinchak - AP Statistics & AP PrecalculusWatch on YouTube (opens in a new tab)

  • Basic Trigonometric Identities: Pythagorean Identities and Cofunction Identities

    Professor Dave ExplainsWatch on YouTube (opens in a new tab)

  • Sum and Difference Identities & Formulas - Sine, Cosine, Tangent - Degrees & Radians, Trigonometry

    The Organic Chemistry TutorWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 3.12 Equivalent Representations of Trigonometric Functions. Pick an answer to see if you got it, and why.

Question 1 of 4

If sin θ = 3/5 and π/2 < θ < π, what is the value of sin(2θ)?

Question 2 of 4

What is the exact value of cos(5π/12)? (Hint: 5π/12 = π/4 + π/6.)

Question 3 of 4

Which of the following is equivalent to cos⁴θ − sin⁴θ?

Question 4 of 4

What are all solutions to 2 sin²θ − cos θ − 1 = 0 for 0 ≤ θ < 2π?

0 of 4 answered