This set has twelve original questions, ordered from easier to harder, covering all five lessons in trigonometric equations and graphs. Questions 1 to 4 are warm-ups, 5 to 8 build the key methods, and 9 to 12 mix skills and ask you to check your count.
Attempt each question on paper, and sketch whenever a question asks for all solutions. Give angles to 1 decimal place unless an exact answer is asked for. Open each answer only after you have tried, mark the ones you got wrong, then use the routing list at the end.
Questions
1. Solve sin x = 0.5 for 0° ≤ x ≤ 360°.
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Sine is positive, so quadrants 1 and 2. The basic angle is 30°. The angles are 30° and 180° − 30° = 150°.
x = 30° and 150°
2. Solve cos x = −0.8 for 0° ≤ x ≤ 360°.
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Cosine is negative, so quadrants 2 and 3. The basic angle is cos⁻¹(0.8) = 36.87°. Quadrant 2: 180° − 36.87° = 143.13°. Quadrant 3: 180° + 36.87° = 216.87°.
x = 143.1° and 216.9°
Check: cos 143.13° ≈ −0.8 and cos 216.87° ≈ −0.8.
3. Solve tan x = 3 for 0° ≤ x ≤ 360°.
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Tangent is positive, so quadrants 1 and 3. The basic angle is tan⁻¹(3) = 71.57°. Quadrant 3: 180° + 71.57° = 251.57°.
x = 71.6° and 251.6°
4. For y = 1 − 4 sin 3x, state the amplitude, the period and the maximum and minimum values.
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Amplitude = |−4| = 4. Period = 360°/3 = 120°. The midline is y = 1, so maximum = 1 + 4 = 5 and minimum = 1 − 4 = −3.
Amplitude 4, period 120°, maximum 5, minimum −3
5. The graph of y = 3 cos(2x − 40°) is drawn for x ≥ 0°. State the period, the horizontal shift and the smallest positive x where y = 3.
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Period: 360°/2 = 180°. Factorise: 2x − 40° = 2(x − 20°), so the shift is 20° to the right. The maximum of 3 occurs when the bracket is 0°, so x = 20°.
Period 180°, shift 20° right, y = 3 first at x = 20°
Check: 3 cos(40° − 40°) = 3 cos 0° = 3.
6. Solve 2 cos²x + cos x − 1 = 0 for 0° ≤ x ≤ 360°.
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Let c = cos x. Then 2c² + c − 1 = 0, which factorises as (2c − 1)(c + 1) = 0, so c = 1/2 or c = −1.
For cos x = 1/2: basic angle 60°, quadrants 1 and 4: x = 60° and 300°. For cos x = −1: x = 180°.
x = 60°, 180°, 300°
7. Solve cos(x − 20°) = 0.3 for 0° ≤ x ≤ 360°.
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Let u = x − 20°, so −20° ≤ u ≤ 340°. The basic angle is cos⁻¹(0.3) = 72.54°, in quadrants 1 and 4: u = 72.54° and 360° − 72.54° = 287.46°. The value −72.54° is below −20°, so it is out. Then x = u + 20°: 92.54° and 307.46°.
x = 92.5° and 307.5°
Check: cos(72.54°) = 0.3 and cos(287.46°) = 0.3.
8. Solve tan 2x = −1 for 0° ≤ x ≤ 180°.
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Let u = 2x, so 0° ≤ u ≤ 360°. Tangent is negative, so quadrants 2 and 4, with basic angle 45°: u = 135° and 315°. Then x = u/2.
x = 67.5° and 157.5°
Check: two cycles of tangent between 0° and 180°, so two solutions.
9. How many solutions does sin 3x = −0.5 have for 0° ≤ x ≤ 180°? Solve it.
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The period is 120°, so the interval holds 1.5 cycles. Let u = 3x, 0° ≤ u ≤ 540°. sin u = −0.5 has basic angle 30° in quadrants 3 and 4: u = 210° and 330°. The next solutions, 570° and 690°, are too large. The sketch shows the half cycle from 360° to 540° is above zero, so no crossing is missed. Then x = u/3.
2 solutions: x = 70° and 110°
10. A curve y = a sin bx + c, with a and b positive, has a maximum value of 7, a minimum value of −3 and a period of 90°. Find a, b and c.
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a = (7 − (−3)) ÷ 2 = 5. c = (7 + (−3)) ÷ 2 = 2. Period 360°/b = 90°, so b = 4.
y = 5 sin 4x + 2
Check: maximum 2 + 5 = 7, minimum 2 − 5 = −3, period 360°/4 = 90°.
11. Solve 2 sin x = √3 for 0 ≤ x ≤ 2π, giving exact answers.
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sin x = √3/2, which has basic angle π/3. Sine is positive, so quadrants 1 and 2: x = π/3 and π − π/3 = 2π/3.
x = π/3 and 2π/3
12. Solve tan 2x = 1 for 0° ≤ x ≤ 360°. State first how many solutions you expect.
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The period of tan 2x is 90°, so there are 4 cycles in 360°, and tangent takes each value once per cycle: expect 4 solutions. Let u = 2x, 0° ≤ u ≤ 720°. tan u = 1 at u = 45°, 225°, 405°, 585°. Then x = u/2.
x = 22.5°, 112.5°, 202.5°, 292.5°
If you got these wrong
Match each mistake to the lesson that repairs it.
- Only one solution, or an angle outside the interval (questions 1, 2, 3, 11): go to solve a trigonometric equation on a stated interval.
- Wrong period or shift (question 5): go to track a phase shift and period.
- Missing solutions after 2x, 3x or a bracket, or a quadratic in cos (questions 6, 7, 8): go to find all roots after a variable substitution.
- The count of answers does not match the sketch (questions 9, 12): go to use a sketch to detect missing solutions.
- Wrong maximum, minimum or amplitude (questions 4, 10): go to interpret an amplitude change separately from translation.
The quadratic structure explorer and the triangle and bearings reasoning board offer more practice in reading graphs and choosing a relationship. Log each slip in the mistake log and retest queue, and retry a fresh version a few days later.
If questions 9 to 12 keep costing you marks, a teacher can go through your written working in online one-to-one Additional Mathematics tuition.