Equation with non-equation preliminary part (sketch/proof/identity)

8 marks Moderate -0.3

It is given that $\beta$ is an angle between $90°$ and $180°$ such that $\sin \beta = a$. Express $\tan^2 \beta - 3 \sin \beta \cos \beta$ in terms of $a$. [3]
Solve the equation $\sin^2 \theta + 2 \cos^2 \theta = 4 \sin \theta + 3$ for $0° < \theta < 360°$. [5]

8 marks Standard +0.3

Solve the equation $3\sin^2 2\theta + 8\cos 2\theta = 0$ for $0° < \theta < 180°$. [5]
\includegraphics{figure_7b} The diagram shows part of the graph of $y = a + \tan bx$, where $x$ is measured in radians and $a$ and $b$ are constants. The curve intersects the $x$-axis at $\left(-\frac{1}{6}\pi, 0\right)$ and the $y$-axis at $(0, \sqrt{3})$. Find the values of $a$ and $b$. [3]

13 marks Moderate -0.3

Solve, for $0° < x < 180°$, the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [7]
In the triangle $ABC$, $AC = 18$ cm, $\angle ABC = 60°$ and $\sin A = \frac{1}{3}$.
1. Use the sine rule to show that $BC = 4\sqrt{3}$. [4]
2. Find the exact value of $\cos A$. [2]

13 marks Moderate -0.3

Solve, for $0° < x < 180°$, the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [7]
In the triangle $ABC$, $AC = 18$ cm, $\angle ABC = 60°$ and $\sin A = \frac{1}{3}$.
1. Use the sine rule to show that $BC = 4\sqrt{3}$. [4]
2. Find the exact value of $\cos A$. [2]

13 marks Moderate -0.3

Solve, for $0° < x < 180°$, the equation $\sin (2x + 50°) = 0.6$, giving your answers to 1 d. p. [7]
In the triangle $ABC$, $AC = 18$ cm, $\angle ABC = 60°$ and $\sin A = \frac{1}{3}$.
1. Use the sine rule to show that $BC = 4\sqrt{3}$. [4]
2. Find the exact value of $\cos A$. [2]

12 marks Standard +0.2

1. Write down the exact values of $\cos \frac{1}{6}\pi$ and $\tan \frac{1}{6}\pi$ (where the angles are in radians). Hence verify that $x = \frac{1}{6}\pi$ is a solution of the equation $$2 \cos x = \tan 2x.$$ [3]
2. Sketch, on a single diagram, the graphs of $y = 2 \cos x$ and $y = \tan 2x$, for $x$ (radians) such that $0 \leqslant x \leqslant \pi$. Hence state, in terms of $\pi$, the other values of $x$ between 0 and $\pi$ satisfying the equation $$2 \cos x = \tan 2x.$$ [4]
1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve $y = \tan x$, the $x$-axis, and the lines $x = 0.1$ and $x = 0.4$. (Values of $x$ are in radians.) [4]
2. State with a reason whether this approximation is an underestimate or an overestimate. [1]

8 marks Moderate -0.8

1. Sketch the graph of $y = 2 \cos x$ for values of $x$ such that $0° \leq x \leq 360°$, indicating the coordinates of any points where the curve meets the axes. [2]
2. Solve the equation $2 \cos x = 0.8$, giving all values of $x$ between $0°$ and $360°$. [3]
Solve the equation $2 \cos x = \sin x$, giving all values of $x$ between $-180°$ and $180°$. [3]

5 marks Moderate -0.8

Show that, when $x$ is an acute angle, $\tan x \sqrt{1 - \sin^2 x} = \sin x$. [2]
Solve $4 \sin^2 y = \sin y$ for $0° \leq y \leq 360°$. [3]

8 marks Standard +0.3

Given that $\sin \theta = 2 - \sqrt{2}$, find the value of $\cos^2 \theta$ in the form $a + b\sqrt{2}$ where $a$ and $b$ are integers. [3]
Find, in terms of $\pi$, all values of $x$ in the interval $0 \leq x < \pi$ for which $$\cos 3x = \frac{\sqrt{3}}{2}.$$ [5]

5 marks Moderate -0.8

Starting with an equilateral triangle, prove that $\cos 30° = \frac{\sqrt{3}}{2}$. [2]
Solve the equation $2 \sin \theta = -1$ for $0 \leqslant \theta \leqslant 2\pi$, giving your answers in terms of $\pi$. [3]

5 marks Easy -1.2

Sketch the graph of $y = \cos x$ for $0° \leqslant x \leqslant 360°$. On the same axes, sketch the graph of $y = \cos 2x$ for $0° \leqslant x \leqslant 360°$. Label each graph clearly. [3]
Solve the equation $\cos 2x = 0.5$ for $0° \leqslant x \leqslant 360°$. [2]

5 marks Moderate -0.8

Sketch the graph of $y = \sin \theta$ for $0 \leqslant \theta \leqslant 2\pi$. [2]
Solve the equation $2 \sin \theta = -1$ for $0 \leqslant \theta \leqslant 2\pi$. Give your answers in the form $k\pi$. [3]

5 marks Moderate -0.8

Sketch the graph of $y = \tan x$ for $0° \leqslant x \leqslant 360°$. [2]
Solve the equation $4 \sin x = 3 \cos x$ for $0° \leqslant x \leqslant 360°$. [3]

9 marks Moderate -0.3

Solve, for $-90° \leqslant \theta < 270°$, the equation, $$\sin(2\theta + 10°) = -0.6$$ giving your answers to one decimal place. [5]
1. A student's attempt at the question "Solve, for $-90° < x < 90°$, the equation $7\tan x = 8\sin x$" is set out below. \begin{align} 7\tan x &= 8\sin x
  7 \times \frac{\sin x}{\cos x} &= 8\sin x
  7\sin x &= 8\sin x \cos x
  7 &= 8\cos x
  \cos x &= \frac{7}{8}
  x &= 29.0° \text{ (to 3 sf)} \end{align} Identify two mistakes made by this student, giving a brief explanation of each mistake. [2]
2. Find the smallest positive solution to the equation $$7\tan(4\alpha + 199°) = 8\sin(4\alpha + 199°)$$ [2]

6 marks Moderate -0.8

Sketch the graph of $y = \cos x$ in the interval $0 \leq x \leq 2\pi$. State the values of the intercepts with the coordinate axes. [2 marks]
1. Given that $$\sin^2 \theta = \cos \theta(2 - \cos \theta)$$ prove that $\cos \theta = \frac{1}{2}$. [2 marks]
2. Hence solve the equation $$\sin^2 2x = \cos 2x(2 - \cos 2x)$$ in the interval $0 \leq x \leq \pi$ [2 marks]

9 marks Standard +0.3

On the same diagram, sketch the graphs of $y = 2 \sec x$ and $y = 1 + 3 \cos x$, for $0 \leqslant x \leqslant \pi$. [4]
Solve the equation $2 \sec x = 1 + 3 \cos x$, where $0 \leqslant x \leqslant \pi$. [5]