Theory question 112
Trigonometry(a) Copy and complete the table of values for y = sin x + 2 cos x, correct to one decimal place.
x: 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°
y: __, 2.2, __, __, __, −1.2, −2.0, __, −1.9
(b) Using a scale of 2 cm to 30° on the x-axis and 2 cm to 0.5 units on the y-axis, draw the graph of y = sin x + 2 cos x for 0° ≤ x ≤ 240°.
(c) Use your graph to solve the equation: (i) sin x + 2 cos x = 0; (ii) sin x = 2.1 − 2 cos x.
(d) From the graph, find y when x = 171°.
Show the model answer
(a) 0°: 2.0; 60°: 1.9; 90°: 1.0; 120°: −0.1; 210°: −2.2 (given: 30°: 2.2, 150°: −1.2, 180°: −2.0, 240°: −1.9).
(b) Plot and join with a smooth curve.
(c)(i) Where the curve crosses the x-axis: x ≈ 117°.
(ii) Rewrite as sin x + 2 cos x = 2.1; draw y = 2.1: x ≈ 7° and x ≈ 47°.
(d) At x = 171°, y ≈ −1.8.
(Accept graph readings close to these.)
For (c)(ii), move 2 cos x to the left so the equation matches the graph.