1.05o Trigonometric equations: solve in given intervals

9 \includegraphics[max width=\textwidth, alt={}, center]{3149080d-ad1a-4d2e-8e20-eb9977ced619-14_558_686_260_726} The diagram shows the curves \(y = \cos x\) and \(y = \frac { k } { 1 + x }\), where \(k\) is a constant, for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = p\).

1. Show that \(p\) satisfies the equation \(\tan p = \frac { 1 } { 1 + p }\).
2. Use the iterative formula \(p _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + p _ { n } } \right)\) to determine the value of \(p\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Hence find the value of \(k\) correct to 2 decimal places.

5 By first expressing the equation $$\tan \theta \tan \left( \theta + 45 ^ { \circ } \right) = 2 \cot 2 \theta$$ as a quadratic equation in \(\tan \theta\), solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

3

1. Given that \(\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)\), show that \(\tan x = \frac { 2 - \sqrt { 3 } } { 1 - 2 \sqrt { 3 } }\).
2. Hence solve the equation $$\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5

1. By first expanding \(\tan ( 2 \theta + 2 \theta )\), show that the equation \(\tan 4 \theta = \frac { 1 } { 2 } \tan \theta\) may be expressed as \(\tan ^ { 4 } \theta + 2 \tan ^ { 2 } \theta - 7 = 0\).
2. Hence solve the equation \(\tan 4 \theta = \frac { 1 } { 2 } \tan \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

3 Solve the equation \(2 \cot 2 x + 3 \cot x = 5\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

2 Solve the equation \(3 \cos 2 \theta = 3 \cos \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

2 Solve the equation \(\cos \left( \theta - 60 ^ { \circ } \right) = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Show that the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) can be expressed in the form $$\cos ^ { 2 } \theta + 2 \sin \theta \cos \theta - 3 \sin ^ { 2 } \theta = 0$$
2. Hence solve the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4 Solve the equation \(2 \cos x - \cos \frac { 1 } { 2 } x = 1\) for \(0 \leqslant x \leqslant 2 \pi\).

6

1. Express \(3 \cos x + 2 \cos \left( x - 60 ^ { \circ } \right)\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$3 \cos 2 \theta + 2 \cos \left( 2 \theta - 60 ^ { \circ } \right) = 2.5$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + 4 y ^ { 2 } } { \mathrm { e } ^ { x } }$$ It is given that \(y = 0\) when \(x = 1\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
2. State what happens to the value of \(y\) as \(x\) tends to infinity.

5 The angles \(\alpha\) and \(\beta\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) and are such that $$\tan ( \alpha + \beta ) = 2 \quad \text { and } \quad \tan \alpha = 3 \tan \beta .$$ Find the possible values of \(\alpha\) and \(\beta\).

6

1. Express \(5 \sin \theta + 12 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. Hence solve the equation \(5 \sin 2 x + 12 \cos 2 x = 6\) for \(0 \leqslant x \leqslant \pi\).

8

1. Express \(3 \sin x + 2 \sqrt { 2 } \cos \left( x + \frac { 1 } { 4 } \pi \right)\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
2. Hence solve the equation $$6 \sin \frac { 1 } { 2 } \theta + 4 \sqrt { 2 } \cos \left( \frac { 1 } { 2 } \theta + \frac { 1 } { 4 } \pi \right) = 3$$ for \(- 4 \pi < \theta < 4 \pi\).

6

1. Express \(\sqrt { 6 } \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(\sqrt { 6 } \cos \frac { 1 } { 3 } x + 3 \sin \frac { 1 } { 3 } x = 2.5\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

4

1. Show that the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 \cot \theta\) can be written in the form $$\tan ^ { 2 } \theta + 3 \sqrt { 3 } \tan \theta - 2 = 0$$
2. Hence solve the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 \cot \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5

1. Show that the equation $$\cot 2 \theta + \cot \theta = 2$$ can be expressed as a quadratic equation in \(\tan \theta\).
2. Hence solve the equation \(\cot 2 \theta + \cot \theta = 2\), for \(0 < \theta < \pi\), giving your answers correct to 3 decimal places.

8

1. By first expanding \(\left( \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) ^ { 2 }\), show that $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta \equiv 1 - \frac { 1 } { 2 } \sin ^ { 2 } 2 \theta .$$
2. Hence solve the equation $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta = \frac { 5 } { 9 } ,$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5 Solve the equation \(\sin \theta = 3 \cos 2 \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4 Solve the equation \(\tan \left( x + 45 ^ { \circ } \right) = 2 \cot x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

6

1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta + 3 \equiv 8 \cos ^ { 4 } \theta\).
2. Hence solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4

1. Express \(4 \cos x - \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(4 \cos 2 x - \sin 2 x = 3\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

7

1. Show that the equation \(\sqrt { 5 } \sec x + \tan x = 4\) can be expressed as \(R \cos ( x + \alpha ) = \sqrt { 5 }\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [4]
2. Hence solve the equation \(\sqrt { 5 } \sec 2 x + \tan 2 x = 4\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

5

1. Given that $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ find the exact value of \(\tan x\).
2. Hence find the exact roots of the equation $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ for \(0 \leqslant x \leqslant 2 \pi\).

7

1. By expressing \(3 \theta\) as \(2 \theta + \theta\), prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
2. Hence solve the equation $$\cos 3 \theta + \cos \theta \cos 2 \theta = \cos ^ { 2 } \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).