Trigonometric curve intersections

Questions involving sketching a trigonometric curve (sine or cosine) and another curve (typically linear) to determine number of intersections or solutions.

2 questions · Moderate -0.2

1.02q Use intersection points: of graphs to solve equations 1.05f Trigonometric function graphs: symmetries and periodicities

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CAIE P1 2023 June Q7

5 marks Standard +0.3

7 A curve has equation \(y = 2 + 3 \sin \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant 4 \pi\).

State greatest and least values of \(y\).
Sketch the curve. \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-09_1127_1219_904_495}
State the number of solutions of the equation $$2 + 3 \sin \frac { 1 } { 2 } x = 5 - 2 x$$ for \(0 \leqslant x \leqslant 4 \pi\).

OCR C2 Q3

6 marks Moderate -0.8

(i) Sketch the curve \(y = \sin x ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
(ii) Sketch on the same diagram the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
(iii) Use your diagram to solve the equation

$$\sin x ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ for \(x\) in the interval \(- 180 \leq x \leq 180\).