LEMMA

Advanced trigonometry

Solve trigonometric equations

ALEKS placement

Solve for the trig function first, then find every angle in the interval.

What this covers

  • Isolate the trig function, treating it as a single unknown.
  • For sine or cosine, a target outside [-1, 1] has no real angle. Check this before using inverse trig.
  • Find the reference angle from that value.
  • Place solutions in every quadrant where the sign matches.
  • Add the period times any integer if a general solution is wanted.
  • Keep the specified endpoints: on [0, 2π), include 0 when it works and leave out 2π.

Worked example

Worked example

Solve 2sin θ - 1 = 0 on [0, 2π).

  1. sinθ=21​Isolate the function.
  2. θ=6π​That is the reference angle where sine is 1/2.
  3. θ=6π​,65π​Sine is positive in quadrants I and II.
Another worked example

Solve tan θ = -1 on [0, 2π).

  1. reference angle=4π​A tangent magnitude of 1 comes from equal-magnitude sine and cosine coordinates.
  2. θ=43π​,47π​Tangent is negative in quadrants II and IV. Its solutions repeat after π.

A common mistake

A mistake Lemma catches

From reference angle=4π​

tan4π​=−1tan43π​=−1

“π/4 is only the reference angle, and tan(π/4) = 1, not −1. Tangent is negative in quadrants II and IV, so θ = π − π/4 = 3π/4 (and 2π − π/4 = 7π/4): tan(3π/4) = −1.”

Try one

Sample problem

Find x in the interval [0, π/2].

sin(x)=21​

Practice trig equations free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this