All topics
0° ≤ x ≤ 360°

08Trigonometric Graphs

Key idea

Know the shape, period and range of y = sin x, y = cos x and y = tan x. Solving equations like sin x = 0.5 across a full interval is about symmetry: find the first solution, then use the curve to locate every other angle in range.

Shapes and ranges

sin x and cos x both have period 360° and range −1 to 1. tan x has period 180° with vertical asymptotes at 90° and 270°.

Two solutions per interval

Within 0°–360°, an equation like sin x = k usually has two solutions. Find the principal value, then use symmetry to find the partner.

Which quadrants?

sin is positive in quadrants 1 and 2; cos in 1 and 4; tan in 1 and 3. A negative value flips you into the other two quadrants.

sin x, cos x: period 360°, range −1 to 1
tan x: period 180°, asymptotes at 90°, 270°
sin(180° − x) = sin x, cos(360° − x) = cos x
Common error — Giving only the calculator's first answer. Each equation has more than one solution in 0°–360° — use the symmetry of the curve.
Worked examples
Example 1 · Positive value

Solve sin x = 0.5 for 0° ≤ x ≤ 360°.

  1. Principal value: x = sin⁻¹(0.5) = 30°
  2. sin is positive in quadrants 1 and 2.
  3. Second solution: 180° − 30° = 150°
Example 2 · Negative value

Solve cos x = −0.4 for 0° ≤ x ≤ 360°.

  1. Reference angle: cos⁻¹(0.4) = 66.4°
  2. cos is negative in quadrants 2 and 3.
  3. x = 180° − 66.4° = 113.6°
  4. x = 180° + 66.4° = 246.4°

Exam-style question bank

5 of 5 shown
Foundations[2 marks]

Solve cos x = 0.5 for 0° ≤ x ≤ 360°.

Foundations[2 marks]

State the range of y = sin x and the period of y = tan x.

Standard[4 marks]

Solve sin x = −0.6 for 0° ≤ x ≤ 360°, to 1 decimal place.

Standard[4 marks]

Solve tan x = 1.5 for 0° ≤ x ≤ 360°, to 1 decimal place.

Challenge[5 marks]

Solve 2 sin x + 1 = 0 for 0° ≤ x ≤ 360°.