These eight original questions follow the trigonometric ratios and graphs chapter. Give each one a serious attempt before opening the answer. Set your calculator to degrees.
Questions
1. Solve sin θ = 0.8 for 0° ≤ θ ≤ 360°.
Answer
Calculator: θ = sin⁻¹ 0.8 = 53.13°, so 53.1°. Sine is positive in quadrants 1 and 2, so the other angle is 180° − 53.13° = 126.9°.
2. Solve cos θ = −0.4 for 0° ≤ θ ≤ 360°.
Answer
Calculator: cos⁻¹(−0.4) = 113.58°, so 113.6°. Cosine is negative in quadrants 2 and 3. The other angle is 360° − 113.58° = 246.42°, so 246.4°.
3. Solve tan θ = −1 for 0° ≤ θ ≤ 360°.
Answer
The reference angle is tan⁻¹ 1 = 45°. Tangent is negative in quadrants 2 and 4. So θ = 180° − 45° = 135° and θ = 360° − 45° = 315°.
4. State the amplitude and period of y = 5 cos 4x.
Answer
Amplitude = 5. Period = 360° ÷ 4 = 90°. The curve completes 4 waves between 0° and 360°.
5. Find the maximum and minimum values of y = 2 sin x + 3.
Answer
The middle line is y = 3 and the amplitude is 2. Maximum = 3 + 2 = 5 and minimum = 3 − 2 = 1.
6. List the key points of y = sin 2x for 0° ≤ x ≤ 360°.
Answer
Period = 360° ÷ 2 = 180°, so there are two waves. The curve crosses zero at 0°, 90°, 180°, 270° and 360°. Maxima of 1 are at 45° and 225°. Minima of −1 are at 135° and 315°.
Check one point: sin(2 × 45°) = sin 90° = 1.
7. The graph y = 3 cos x meets the line y = 1.5. Find the values of x for 0° ≤ x ≤ 360°.
Answer
Set 3 cos x = 1.5, so cos x = 0.5. Calculator: x = 60°. The other angle is 360° − 60° = 300°.
8. A student says y = sin 3x has a period of 1 080°. Explain the error and give the correct period.
Answer
The student multiplied 360° by 3. The period is 360° ÷ 3 = 120°.
A sense check agrees: multiplying x by 3 makes the wave repeat three times faster, so the period is shorter than 360°.
If you got these wrong
If questions 1 to 3 went wrong, revisit trigonometric ratios beyond acute angles and finding angles from a trigonometric graph. For questions 4 to 8, see amplitude, period and simple transformations.
Record each slip in the mistake log and paper-error review, then build a timed set with the timed original practice session builder.
For a teacher to go through your working, see online one-to-one Mathematics tuition.