doing math

courses › Trigonometry

Identities & equations

level 43 course

Use trig identities and solve trig equations.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 43.

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Builds on: Radians & the unit circle (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

An identity is an equation that holds for every angle. The main one: the point (cos θ, sin θ) on the unit circle is 1 from the center, so sin² θ + cos² θ = 1. Know one value and the other follows, up to a sign the quadrant decides.

Two more: sin 2θ = 2 sin θ cos θ, and cos 2θ = cos² θ − sin² θ.

Sine and cosine repeat every 2π, tangent every π. That repeat length is the period. Squeezing the input, as in sin(2x), shortens it.

Techniques

Use sin² + cos² = 1

You know sin θ or cos θ and the quadrant, and want the other.

  1. Square the value you know and take it from 1.
  2. Take the square root. That gives the size.
  3. Pick the sign from the quadrant: cosine is negative in II and III, sine in III and IV.
worked example

Example: cos θ = 5/13 and θ is in quadrant IV. What is sin θ? Give a fraction.

  1. sin² θ = 1 − 25/169 = 144/169.
  2. The square root gives a size of 12/13.
  3. Sine is negative in quadrant IV, so sin θ is −12/13.

Answer: −12/13

Period: divide by b

The period of y = sin(bx), cos(bx) or tan(bx).

  1. Sine and cosine: the period is 2π ÷ b. Tangent: π ÷ b.
  2. For k in kπ, divide 2 by b, or 1 by b for tangent.
worked example

Example: The period of y = cos(x/3) is kπ. What is k? Give a fraction or a whole number.

  1. x/3 means b is 1/3.
  2. Cosine: the period is 2π ÷ (1/3).
  3. k = 2 ÷ (1/3) = 6.

Answer: 6

Reference angle, two quadrants

Solving sin x or cos x equal to a special value, for 0° ≤ x < 360°.

  1. Find the reference angle r from the value's size, ignoring its sign.
  2. Find the two quadrants where the function has the value's sign.
  3. Quadrant I gives r, II gives 180° − r, III gives 180° + r, and IV gives 360° − r.
worked example

Example: Solve cos x = −1/2 for 0° ≤ x < 360°. What is the larger answer, in degrees?

  1. cos 60° = 1/2, so the reference angle is 60°.
  2. Cosine is negative in quadrants II and III.
  3. II: 180° − 60° = 120°. III: 180° + 60° = 240°.

Answer: 240

Tips by skill

  • Tipsin² + cos² = 1: Square the known value, take it from 1, and take the square root. The quadrant decides the sign.
  • TipPeriod: Sine and cosine: 2π ÷ b. Tangent: π ÷ b. For x/3, b is 1/3.
  • TipDouble angle: sin 2θ = 2 sin θ cos θ, and cos 2θ = cos² θ − sin² θ. Keep each value's sign.
  • TipSolve a trig equation: Find the reference angle from the value's size, then the two quadrants with the right sign. Enter the smaller angle first.

Watch out for

  • Forgetting the sign. The identity gives the size; the quadrant decides the sign.
  • Doubling sin θ to get sin 2θ. The formula is 2 sin θ cos θ.
  • Multiplying by b instead of dividing. sin(4x) repeats faster, so its period is shorter.
  • Using the other function's quadrants: sine is positive in I and II, cosine in I and IV.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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