Scottish National 5 Mathematics

National 5 Maths Trigonometric equations

Solve sine, cosine and tangent equations in degrees over stated intervals.

Formula referenceMixed revisionPrintable worksheet

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Use inverse trig functions to find a reference angle.
  • Know the signs of sine, cosine and tangent in each quadrant.
  • Check every solution lies in the stated interval.
National 5 Mathematics lesson

Explanation

A trigonometric equation asks for every angle in an interval that gives a stated trig value.

First use an inverse trig function to find a reference angle. Then use the trig graph or quadrant signs to locate all other solutions.

Sine is positive in quadrants 1 and 2, cosine in quadrants 1 and 4, and tangent in quadrants 1 and 3. Negative values use the other quadrants.

Key formulae and rules

  • sin positive: Q1 and Q2
  • cos positive: Q1 and Q4
  • tan positive: Q1 and Q3
  • solutions must be inside the stated interval

Watch out

Giving only the calculator's first solution.

Check

Substitute every solution back into the equation and keep only angles in the stated interval.

Exam tip

Write the reference angle and identify the valid quadrants before listing solutions.

Calculator tip

Use degree mode and keep full calculator values until the final requested rounding.

Worked examples

Worked example 1

Solve a sine equation

Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

  1. Reference angle = sin⁻¹(0.5) = 30°
  2. Sine is positive in quadrants 1 and 2.
  3. The second solution is 180° − 30°.

Answer: x = 30° or 150°

Worked example 2

Solve a cosine equation

Solve cos x = 0.4 for 0° ≤ x ≤ 360°.

  1. Reference angle = cos⁻¹(0.4) = 66.4°
  2. Cosine is positive in quadrants 1 and 4.
  3. The second solution is 360° − 66.4°.

So: x = 66.4° or 293.6°

Worked example 3

Solve a negative sine equation

Solve sin x = −0.6 for 0° ≤ x ≤ 360°.

  1. Reference angle = sin⁻¹(0.6) = 36.9°
  2. Sine is negative in quadrants 3 and 4.
  3. Use 180° + 36.9° and 360° − 36.9°.

Answer: x = 216.9° or 323.1°

Watch out

  • Giving only the calculator's first solution.
  • Using the wrong quadrants for a negative value.
  • Including a solution outside the interval.