Identities & equations
Use trig identities and solve trig equations.
Pen and paper is fine · no calculator needed why?
Opens at level 43. You're level 1. You can read and practice here now.
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
- Square the value you know and take it from 1.
- Take the square root. That gives the size.
- Pick the sign from the quadrant: cosine is negative in II and III, sine in III and IV.
worked example
cos θ = 5/13 and θ is in quadrant IV. What is sin θ? Give a fraction.
- sin² θ = 1 − 25/169 = 144/169.
- The square root gives a size of 12/13.
- Sine is negative in quadrant IV, so sin θ is −12/13.
Answer: −12/13
Period: divide by b
- Sine and cosine: the period is 2π ÷ b. Tangent: π ÷ b.
- For k in kπ, divide 2 by b, or 1 by b for tangent.
worked example
The period of y = cos(x/3) is kπ. What is k? Give a fraction or a whole number.
- x/3 means b is 1/3.
- Cosine: the period is 2π ÷ (1/3).
- k = 2 ÷ (1/3) = 6.
Answer: 6
Reference angle, two quadrants
- Find the reference angle r from the value's size, ignoring its sign.
- Find the two quadrants where the function has the value's sign.
- Quadrant I gives r, II gives 180° − r, III gives 180° + r, and IV gives 360° − r.
worked example
Solve cos x = −1/2 for 0° ≤ x < 360°. What is the larger answer, in degrees?
- cos 60° = 1/2, so the reference angle is 60°.
- Cosine is negative in quadrants II and III.
- II: 180° − 60° = 120°. III: 180° + 60° = 240°.
Answer: 240
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.
practice
sin² + cos² = 1
worked example
sin θ = −20/29 and θ is in quadrant IV. What is cos θ? Give a fraction.
Answer: 21/29
- cos² θ = 1 − sin² θ = 1 − 400/841 = 441/841, so cos θ = ±21/29.
- Cosine is positive in quadrant IV, so cos θ = 21/29.
Period
worked example
The period of y = sin(2x) is kπ. What is k? Give a fraction or a whole number.
Answer: 1
- The period of sin(bx) is 2π ÷ b = 2π ÷ 2 = π, so k = 1.
Double angle
worked example
sin θ = −9/41 and cos θ = −40/41. What is cos 2θ? Give a fraction.
Answer: 1519/1681
- cos 2θ = cos² θ − sin² θ = 1600/1681 − 81/1681 = 1519/1681.
Solve a trig equation
worked example
Solve sin x = 1/2 for 0° ≤ x < 360°. Enter both answers in degrees, smaller first.
Answer: (30, 150)
- The reference angle is 30°, because sin 30° = 1/2.
- Sine is positive in quadrants I and II: 30° and 180° − 30° = 150°.