Standard trigonometric equations

Find all values of \(x\) in the interval \(0 \leq x < 360°\) for which $$2\sin^2 x - 2\cos x - \cos^2 x = 1.$$ [8]

Solve the following equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$ a) Given that \(y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\). b) Find \(\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x\). The diagram below shows a sketch of \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of \(y = 4 + f ( x )\), clearly indicating any asymptotes.
b) Sketch the graph of \(y = f ( x - 3 )\), clearly indicating any asymptotes.
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0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve \(C\) with equation \(y = 14 + 5 x - x ^ { 2 }\) and line \(L\) with equation \(y = x + 2\). The line intersects the curve at the points \(A\) and \(B\).
a) Find the coordinates of \(A\) and \(B\).
b) Calculate the area enclosed by \(L\) and \(C\). Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$

Find all values of \(x\) in the interval \(0 \leq x < 360°\) for which $$2\sin^2 x - 2\cos x - \cos^2 x = 1.$$ [8]

Solve the equation \(2\sin 2\theta + \cos 2\theta = 1\), for \(0° \leqslant \theta < 360°\). [6]

3 Solve the equation $$\cos \theta + 4 \cos 2 \theta = 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

The diagram shows a sketch of part of the curve with equation \(y = 2\sin x + 3\cos^2 x - 3\). The curve crosses the \(x\)-axis at the points O, A, B and C. \includegraphics{figure_15} Find the value of \(x\) at each of the points A, B and C. [7]

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

1. 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]
2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leqslant x \leqslant 360°\). [2]

1. Solve the equation \(3\sin^2 2\theta + 8\cos 2\theta = 0\) for \(0° < \theta < 180°\). [5]
2. \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]

Solve the equation \(\tan(\theta - 60°) = 3 \cot \theta\) for \(-90° < \theta < 90°\). [5]

11

1. Solve the equation \(3 \tan ^ { 2 } x - 5 \tan x - 2 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
2. Find the set of values of \(k\) for which the equation \(3 \tan ^ { 2 } x - 5 \tan x + k = 0\) has no solutions.
3. For the equation \(3 \tan ^ { 2 } x - 5 \tan x + k = 0\), state the value of \(k\) for which there are three solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), and find these solutions.

In this question you must show detailed reasoning. Solve the following equation for \(x\) in the interval \(0° < x < 180°\) $$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]

Solve the equation \(\tan 2\theta = 3\) for \(0° < \theta < 360°\). [3]

4 Solve the equation \(\tan 2 \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
[0pt] [3]

1 Solve the equation \(\sin 2 x = 2 \cos 2 x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

Solve each of the following equations, for \(0° < x < 360°\).

1. \(\sin \frac{1}{2}x = 0.8\) [3]
2. \(\sin x = 3 \cos x\) [3]

Solve, for \(0 \leq x \leq 180°\), the equation $$\sin(x + 10°) = \frac{\sqrt{3}}{2}.$$ [4]

1. \(\cos 2x = -0.9\), giving your answers to 1 decimal place. [4]

Find all the values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which

1. \(\cos(\theta - 10°) = \cos 15°\), [3]
2. \(\tan 2\theta = 0.4\), [5]
3. \(2 \sin \theta \tan \theta = 3\). [6]

Find the general solution of the equation $$\cos 4\theta + \cos 2\theta = \cos\theta.$$ [6]

3 Find the general solution of the equation $$\sin \left( 4 x + \frac { \pi } { 4 } \right) = 1$$

Find the general solution of the equation $$\sin 6\theta + 2\cos^2\theta = 3\cos 2\theta - \sin 2\theta + 1.$$ [9]

In this question you must show detailed reasoning.

1. Show that \(\cos A + \sin A \tan A = \sec A\). [3]
2. Solve the equation \(\tan 2\theta = 3 \tan \theta\) for \(0° \leqslant \theta \leqslant 180°\). [7]

4

1. Prove that \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } \equiv 2 \tan \theta\). \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_63_1569_333_328} .............................................................................................................................................. \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_65_1570_511_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_62_1570_603_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_685_324} \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_776_324} ...................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_76_1572_952_322} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_1137_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_74_1572_1226_322} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_77_1575_1315_319}
2. Hence solve the equation \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } = 5 \tan ^ { 3 } \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

10

1. Show that \(\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta\).
2. Hence solve the equation \(\sin 3 \theta = \cos 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
3. Show that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\). Hence, by writing \(\cos 2 \theta - \sin 3 \theta\) in terms of \(\sin \theta\), use your answer to part (ii) to determine the solutions of \(4 x ^ { 3 } - 2 x ^ { 2 } - 3 x + 1 = 0\).

1 Solve the equation \(2 \cos \theta = 7 - \frac { 3 } { \cos \theta }\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

1. Show that the equation \(\frac{\tan \theta}{\cos \theta} = 1\) may be rewritten as \(\sin \theta = 1 - \sin^2 \theta\). [2]
2. Hence solve the equation \(\frac{\tan \theta}{\cos \theta} = 1\) for \(0° \leq \theta \leq 360°\). [3]

Solve for \(0 < x < 360°\) $$\cot 2x - \tan 78° = \frac{(\sec x)(\sec 78°)}{2}$$ where \(x\) is not an integer multiple of \(90°\) [9]

1. Given that $$5 \cos \theta - 2 \sin \theta = 0,$$ show that \(\tan \theta = 2.5\) [2]
2. Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2x° - 2 \sin 2x° = 0,$$ giving your answers to 1 decimal place. [4]

1. Given that $$5 \cos \theta - 2 \sin \theta = 0,$$ show that \(\tan \theta = 2.5\) [2]
2. Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2x° - 2 \sin 2x° = 0,$$ giving your answers to 1 decimal place. [4]

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Show that the equation
   $$\sin \theta ( 7 \sin \theta - 4 \cos \theta ) = 4$$ can be written as $$3 \tan ^ { 2 } \theta - 4 \tan \theta - 4 = 0$$
2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$\sin x ( 7 \sin x - 4 \cos x ) = 4$$ giving your answers to one decimal place.
3. Hence find the smallest solution of the equation $$\sin 4 \alpha ( 7 \sin 4 \alpha - 4 \cos 4 \alpha ) = 4$$ in the range \(720 ^ { \circ } < \alpha < 1080 ^ { \circ }\), giving your answer to one decimal place.

Solve the equation \(\tan^2 2\theta - 3 = 0\) giving all the solutions for \(0° \leq \theta \leq 360°\) [4 marks]

Solve \(2 \sin x = \tan x\) exactly, where \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). [4]

Solve for \(0 \leq \theta \leq 180°\) $$\tan(\theta + 35°) = \cot(\theta - 53°)$$ [Total 4 marks]

Solve the equation $$\cos(x + 30°) = 2\cos x,$$ giving all solutions in the interval \(-180° < x < 180°\). [5]

Simplify \(\frac{\sqrt{1 - \cos^2 \theta}}{\tan \theta}\), where \(\theta\) is an acute angle. [3]

Solve the equation $$\cos 2\theta - \cos 4\theta = \sin 3\theta \quad \text{for} \quad 0 \leq \theta \leq \pi$$ [6]

5

1. Find the general solution of the equation $$\cos ( 3 x - \pi ) = \frac { 1 } { 2 }$$ giving your answer in terms of \(\pi\).
2. From your general solution, find all the solutions of the equation which lie between \(10 \pi\) and \(11 \pi\).

1. Solve the equation \(\text{p}(\cos ec^2 \theta) = 0\) for \(-90° < \theta < 90°\). [3]

8. (i) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), $$\tan \left( x - 40 ^ { \circ } \right) = 1.5$$ giving your answers to 1 decimal place.
(ii) (a) Show that the equation $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ can be written in the form $$4 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ showing each stage of your working.

Solve the equation $$\sin^2 x = 1$$ for \(0° < x < 360°\) [3 marks]

Sketch the curve \(y = \sin x\) for \(0° \leqslant x \leqslant 360°\). Solve the equation \(\sin x = -0.68\) for \(0° \leqslant x \leqslant 360°\). [4]

1. Given that \(\frac{\sin \theta - \cos \theta}{\cos \theta} = 4\), prove that \(\tan \theta = 5\). [2]
2. 1. Use an appropriate identity to show that the equation $$2 \cos^2 x - \sin x = 1$$ can be written as $$2 \sin^2 x + \sin x - 1 = 0$$ [2]
   2. Hence solve the equation $$2 \cos^2 x - \sin x = 1$$ giving all solutions in the interval \(0° \leq x \leq 360°\). [5]

Use an isosceles right-angled triangle to show that \(\cos 45° = \frac{1}{\sqrt{2}}\). [2]

4 The equation of a curve is \(y = 2 x - \tan x\), where \(x\) is in radians. Find the coordinates of the stationary points of the curve for which \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).