1.05o Trigonometric equations: solve in given intervals

1. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \tan ^ { 2 } \theta \equiv \sec ^ { 2 } \theta\).
   (b) Solve, for \(0 \leq \theta < 360 ^ { \circ }\), the equation

$$2 \tan ^ { 2 } \theta + \sec \theta = 1 ,$$ giving your answers to 1 decimal place.

5. (a) Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$ (b) Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$ (c) Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(d) Hence, for \(0 \leq \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
Hence, for \(0 \leq \theta < \pi\), solve \includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.

8

1. Solve the equation \(\cos x = 0.3\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
2. The diagram shows the graph of \(y = \cos x\) for \(0 \leqslant x \leqslant 2 \pi\) and the line \(y = k\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-5_524_805_559_648} The line \(y = k\) intersects the curve \(y = \cos x , 0 \leqslant x \leqslant 2 \pi\), at the points \(P\) and \(Q\). The point \(M\) is the minimum point of the curve.
   1. Write down the coordinates of the point \(M\).
   2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of \(Q\) in terms of \(\pi\) and \(\alpha\).
3. Describe the geometrical transformation that maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
4. Solve the equation \(\cos 2 x = \cos \frac { 4 \pi } { 5 }\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving the values of \(x\) in terms of \(\pi\).
   (4 marks)

7

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Write down the two solutions of the equation \(\tan x = \tan 61 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. 1. Given that \(\sin \theta + \cos \theta = 0\), show that \(\tan \theta = - 1\).
   2. Hence solve the equation \(\sin \left( x - 20 ^ { \circ } \right) + \cos \left( x - 20 ^ { \circ } \right) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
4. Describe the single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x - 20 ^ { \circ } \right)\).
5. The curve \(y = \tan x\) is stretched in the \(x\)-direction with scale factor \(\frac { 1 } { 4 }\) to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

1. Giving your answers in terms of \(\pi\), solve the equation

$$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).

4 It is given that \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\).

1. Show that the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\) can be written in the form $$2 \cot ^ { 2 } x + 5 \cot x - 3 = 0$$
2. Hence show that \(\tan x = 2\) or \(\tan x = - \frac { 1 } { 3 }\).
3. Hence, or otherwise, solve the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\), giving all values of \(x\) in radians to one decimal place in the interval \(- \pi < x \leqslant \pi\).

4

1. Solve the equation \(\sec x = \frac { 3 } { 2 }\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. By using a suitable trigonometrical identity, solve the equation $$2 \tan ^ { 2 } x = 10 - 5 \sec x$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3

1. Solve the equation $$\operatorname { cosec } x = 3$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (2 marks)
2. By using a suitable trigonometric identity, solve the equation $$\cot ^ { 2 } x = 11 - \operatorname { cosec } x$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (6 marks)

3

1. Solve the equation \(\operatorname { cosec } x = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)
2. The diagram shows the graph of \(y = \operatorname { cosec } x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-03_609_1045_559_479}
   1. The point \(A\) on the curve is where \(x = 90 ^ { \circ }\). State the \(y\)-coordinate of \(A\).
   2. Sketch the graph of \(y = | \operatorname { cosec } x |\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Solve the equation \(| \operatorname { cosec } x | = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)

8

1. Show that the equation \(4 \operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta = k\), where \(k \neq 4\), can be written in the form $$\sec ^ { 2 } \theta = \frac { k - 1 } { k - 4 }$$
2. Hence, or otherwise, solve the equation $$4 \operatorname { cosec } ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) - \cot ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) = 5$$ giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
   [0pt] [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-18_72_113_1055_159}
   
   \includegraphics[max width=\textwidth, alt={}]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-20_2288_1707_221_153}

3

1. Express \(\cos 2 x\) in terms of \(\sin x\).
2. 1. Hence show that \(3 \sin x - \cos 2 x = 2 \sin ^ { 2 } x + 3 \sin x - 1\) for all values of \(x\).
   2. Solve the equation \(3 \sin x - \cos 2 x = 1\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Use your answer from part (a) to find \(\int \sin ^ { 2 } x \mathrm {~d} x\).

7

1. 1. Express \(6 \sin \theta + 8 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(6 \sin 2 x + 8 \cos 2 x = 7\), giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. 1. Prove the identity \(\frac { \sin 2 x } { 1 - \cos 2 x } = \frac { 1 } { \tan x }\).
   2. Hence solve the equation $$\frac { \sin 2 x } { 1 - \cos 2 x } = \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

2

1. Express \(\sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) in radians to two decimal places.
2. Hence:
   1. write down the minimum value of \(\sin x - 3 \cos x\);
   2. find the value of \(x\) in the interval \(0 < x < 2 \pi\) at which this minimum value occurs, giving your value of \(x\) in radians to two decimal places.

5

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Solve the equation $$5 \sin 2 x + 3 \cos x = 0$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) to the nearest \(0.1 ^ { \circ }\), where appropriate.
3. Given that \(\sin 2 x + \cos 2 x = 1 + \sin x\) and \(\sin x \neq 0\), show that \(2 ( \cos x - \sin x ) = 1\).

2

1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\), in radians, to three decimal places.
2. 1. Hence write down the minimum value of \(\cos x + 3 \sin x\).
   2. Find the value of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) at which this minimum occurs, giving your answer, in radians, to three decimal places.
3. Solve the equation \(\cos x + 3 \sin x = 2\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving all solutions, in radians, to three decimal places.

1

1. Express \(2 \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R\) is a positive constant and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Solve the equation \(2 \sin x + \cos x = 1\) for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).

4

1. 1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
   2. Express \(\cos 2 x\) in terms of \(\cos x\).
2. Show that $$\sin 2 x - \tan x = \tan x \cos 2 x$$ for all values of \(x\).
3. Solve the equation \(\sin 2 x - \tan x = 0\), giving all solutions in degrees in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3

1. Express \(4 \cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 360 ^ { \circ }\), giving your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Hence solve the equation \(4 \cos x + 3 \sin x = 2\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving all solutions to the nearest \(0.1 ^ { \circ }\).
3. Write down the minimum value of \(4 \cos x + 3 \sin x\) and find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) at which this minimum value occurs.

6

1. 1. Show that the equation \(3 \cos 2 x + 7 \cos x + 5 = 0\) can be written in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers.
   2. Hence find the possible values of \(\cos x\).
2. 1. Express \(7 \sin \theta + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(7 \sin \theta + 3 \cos \theta = 4\) for all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving \(\theta\) to the nearest \(0.1 ^ { \circ }\).
3. 1. Given that \(\beta\) is an acute angle and that \(\tan \beta = 2 \sqrt { 2 }\), show that \(\cos \beta = \frac { 1 } { 3 }\).
   2. Hence show that \(\sin 2 \beta = p \sqrt { 2 }\), where \(p\) is a rational number.

4 Solve the equation \(2 \sin 2 \theta + \cos 2 \theta = 1\), for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

3 Given that \(\sin ( \theta + \alpha ) = 2 \sin \theta\), show that \(\tan \theta = \frac { \sin \alpha } { 2 - \cos \alpha }\). Hence solve the equation \(\sin \left( \theta + 40 ^ { \circ } \right) = 2 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).

6 The angle \(x\) radians satisfies the equation $$\cos \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { \sqrt { 2 } }$$

1. Find the general solution of this equation, giving the roots as exact values in terms of \(\pi\).
2. Find the number of roots of the equation which lie between 0 and \(2 \pi\).

3 Find the general solution of the equation $$\tan 4 \left( x - \frac { \pi } { 8 } \right) = 1$$ giving your answer in terms of \(\pi\).

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