1.05o Trigonometric equations: solve in given intervals

7

1. Prove the identity \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 }\).
2. Hence find the exact solutions of the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 2\) for \(0 \leqslant \theta \leqslant \pi\).

1 Solve the equation \(8 \sin ^ { 2 } \theta + 6 \cos \theta + 1 = 0\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5

1. Show that the equation $$4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0$$ may be expressed in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers to be found.
2. Hence solve the equation \(4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

3

1. Show that the equation $$5 \cos \theta - \sin \theta \tan \theta + 1 = 0$$ may be expressed in the form \(a \cos ^ { 2 } \theta + b \cos \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be found.
2. Hence solve the equation \(5 \cos \theta - \sin \theta \tan \theta + 1 = 0\) for \(0 < \theta < 2 \pi\).

7

1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

2

1. Show that \(\sin x \tan x\) may be written as \(\frac { 1 - \cos ^ { 2 } x } { \cos x }\).
2. Hence solve the equation \(2 \sin x \tan x = 3\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6 The function f , where \(\mathrm { f } ( x ) = a \sin x + b\), is defined for the domain \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 2\) and that \(\mathrm { f } \left( \frac { 3 } { 2 } \pi \right) = - 8\),

1. find the values of \(a\) and \(b\),
2. find the values of \(x\) for which \(\mathrm { f } ( x ) = 0\), giving your answers in radians correct to 2 decimal places,
3. sketch the graph of \(y = \mathrm { f } ( x )\).

2 Find all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) which satisfy the equation \(\sin 3 x + 2 \cos 3 x = 0\).

3

1. Show that the equation \(\sin ^ { 2 } \theta + 3 \sin \theta \cos \theta = 4 \cos ^ { 2 } \theta\) can be written as a quadratic equation in \(\tan \theta\).
2. Hence, or otherwise, solve the equation in part (i) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

3

1. Show that the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\) can be expressed as \(\tan \theta = 3\).
2. Hence solve the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

2 Solve the equation $$\sin 2 x + 3 \cos 2 x = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

8 The function f is defined by \(\mathrm { f } ( x ) = a + b \cos 2 x\), for \(0 \leqslant x \leqslant \pi\). It is given that \(\mathrm { f } ( 0 ) = - 1\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 7\).

1. Find the values of \(a\) and \(b\).
2. Find the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).

2

1. Show that the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\) can be written in the form \(2 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0\).
2. Hence solve the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Show that the equation \(2 \sin x \tan x + 3 = 0\) can be expressed as \(2 \cos ^ { 2 } x - 3 \cos x - 2 = 0\).
2. Solve the equation \(2 \sin x \tan x + 3 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5

1. Show that the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) can be written in the form $$2 \sin ^ { 4 } \theta + \sin ^ { 2 } \theta - 1 = 0 .$$
2. Hence solve the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

8

1. Prove the identity \(\left( \frac { 1 } { \sin \theta } - \frac { 1 } { \tan \theta } \right) ^ { 2 } \equiv \frac { 1 - \cos \theta } { 1 + \cos \theta }\).
2. Hence solve the equation \(\left( \frac { 1 } { \sin \theta } - \frac { 1 } { \tan \theta } \right) ^ { 2 } = \frac { 2 } { 5 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

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

5

1. Show that \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { 1 } { \sin ^ { 2 } \theta - \cos ^ { 2 } \theta }\).
2. Hence solve the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 3\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3

1. Express the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) as a quadratic equation in \(\cos ^ { 2 } \theta\).
2. Solve the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 \leqslant \theta \leqslant \pi\), giving solutions in terms of \(\pi\).

9

1. Prove the identity \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } \equiv \frac { 1 } { \tan \theta }\).
2. Hence solve the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 4 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5

1. Prove the identity \(\frac { 1 } { \cos \theta } - \frac { \cos \theta } { 1 + \sin \theta } \equiv \tan \theta\).
2. Solve the equation \(\frac { 1 } { \cos \theta } - \frac { \cos \theta } { 1 + \sin \theta } + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Prove the identity \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } \equiv \sin x + \cos x\).
2. Hence solve the equation \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } = 3 \sin x - 2 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).

4

1. Express the equation \(3 \sin \theta = \cos \theta\) in the form \(\tan \theta = k\) and solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
2. Solve the equation \(3 \sin ^ { 2 } 2 x = \cos ^ { 2 } 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

2 Solve the equation \(3 \sin ^ { 2 } \theta = 4 \cos \theta - 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

7

1. Prove the identity \(\frac { 1 + \cos \theta } { 1 - \cos \theta } - \frac { 1 - \cos \theta } { 1 + \cos \theta } \equiv \frac { 4 } { \sin \theta \tan \theta }\).
2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$\sin \theta \left( \frac { 1 + \cos \theta } { 1 - \cos \theta } - \frac { 1 - \cos \theta } { 1 + \cos \theta } \right) = 3 .$$