About this chapter
Trigonometric ratios brings together two very different kinds of question. Triangle problems reward careful reading, such as noticing that an angle has to be obtuse, and keeping full accuracy until the end. Graph problems reward understanding: the period of tan x, where a transformed graph crosses the axes, and how many times a line meets a curve over a long interval.
This chapter works in degrees; radians follow in Chapter 7. It covers the cosine rule for sides and angles, the sine rule and its ambiguous case, ½ab sin C, and the graphs of sine, cosine and tangent with their transformations. The hardest questions count the solutions of tan x = x/45 up to ±9000° and use the cosine rule as a quadratic with surd roots.
The ten questions
- Q1 (4 marks): the third side of a triangle by the cosine rule, then its area.
- Q2 (5 marks): the two possible angles from the sine rule, then a side when the angle is obtuse.
- Q3 (5 marks): a 5, 7, 9 triangle: show that cos θ = −1/10 and find the exact area in surd form.
- Q4 (5 marks): sketch y = tan(x − 45°) with its asymptotes, then count solutions of two equations.
- Q5 (5 marks): read a and b in f(x) = a cos bx from its graph, then its minimum points and a solution count.
- Q6 (8 marks): a four-sided garden: the cosine rule, a sine rule angle with only one possible value, and the total area.
- Q7 (9 marks): the curve y = −4cos 2x: its key points, a vertical translation through a given point, and a drawn line to count solutions.
- Q8 (8 marks): y = 2 sin(x + 60°): exact axis crossings, turning points, when y = k meets it twice, and a related equation.
- Q9 (8 marks): the period of tan x, then how many times tan x = x/45 holds over wider and wider intervals.
- Q10 (9 marks): an obtuse angle from the sine rule, the cosine rule as a quadratic giving AC = 5√3 − √11, then a second triangle.
Key skills tested
Cosine rule for a side. a² = b² + c² − 2bc cos A, used with two sides and the angle between them.
Cosine rule for an angle. Rearrange to cos A = (b² + c² − a²)/2bc. A negative value means the angle is obtuse.
Sine rule. a/sin A = b/sin B = c/sin C, used when you know a side and its opposite angle.
The ambiguous case. sin θ = sin(180° − θ), so an angle from the sine rule may be acute or obtuse. Check whether both fit in the triangle.
Area of a triangle. ½ab sin C, where C is the angle between sides a and b. Given the area, sin C may lead to two possible angles.
Multi-step triangle problems. Split the shape into triangles, find the shared side first, and keep full calculator accuracy until the final answer.
Graphs of sin, cos and tan. Sine and cosine have period 360° and stay between −1 and 1; tan has period 180° with asymptotes at 90° and 270°.
Transforming trig graphs. y = a sin x has height a; y = sin(x + 30°) moves 30° to the left; y = sin bx has period 360°/b.
Counting solutions. Sketch both graphs over the whole interval and count the intersections, checking end points, asymptotes and repeated periods.
Worked example
Question 1 from this chapter. In triangle ABC, AB = 8 cm, AC = 11 cm and angle BAC = 50°. Find (a) the length BC and (b) the area of the triangle, both to 3 significant figures.
(a) Two sides and the angle between them call for the cosine rule: BC² = 8² + 11² − 2 × 8 × 11 × cos 50° = 185 − 176 cos 50° = 71.87 to 4 significant figures. So BC = 8.48 cm.
(b) The area is ½ × 8 × 11 × sin 50° = 44 sin 50° = 33.7 cm².
Keep BC² unrounded in your calculator until the last step. Rounding it early can move the third significant figure of BC, and that costs the accuracy mark.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2025, Q8: an obtuse angle from the sine rule, a triangle's area and a composite perimeter, the triangle skills of Q2, Q6 and Q10.
- June 2024, Q5: the cosine rule for an angle and the areas of two triangles, as in Q6.
- October 2024, Q7: key points on y = −4cos 2x, a vertical translation, then a line drawn on the figure to count solutions, as in Q7.
- January 2023, Q9: the period of tan x and the number of roots of an equation over growing intervals, as in Q9.
- June 2023, Q9: identifying a trig function from its graph, then sketching a transformation of it, as in Q5 and Q8.
Where marks are lost
- Missing the word obtuse. In January 2025 a large share of students either missed that the question asked for an obtuse angle or did not notice their angle was acute, and the error carried into every later part.
- Rounding too early. In June 2024 too many lost the final accuracy mark through rounding or truncating intermediate values.
- The period of tan x. In January 2023 fewer than half gave the period correctly; many wrote an interval such as −π/2 < x < π/2 instead of a single value.
- A line with no conclusion. When asked to draw a line on a graph and count solutions in October 2024, many stated a number that did not match their own line, or gave no reason.
- sin 2x or sin(x/2)? In June 2023 a common error was reading a graph that repeats twice as often as sin(x/2) instead of sin 2x.
Common questions
Are the sine and cosine rules in the formula booklet?
No. The sine rule, the cosine rule and ½ab sin C all have to be learned for WMA11.
How do I know if there are two possible angles?
An angle found with the sine rule could also be 180° minus that angle. Both are possible only if the obtuse one still leaves room for the other angles in the triangle.
Degrees or radians in this chapter?
Degrees throughout. Chapter 7 repeats the graph work in radians, which is how the real paper often sets it.
Where this chapter leads
- Pure 1 Chapter 7: Radians: arcs, sectors and trig graphs with angles in radians.
- Pure 2 Chapter 6: Trigonometric Identities and Equations: solving trig equations over an interval.