1.05o Trigonometric equations: solve in given intervals

7. (i) Use the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$ (ii) Prove that, for \(\sin x \neq 0\), $$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$ (iii) Find the values of \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) for which $$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$ giving your answers to 1 decimal place where appropriate.

1 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).

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

5 Solve the equation \(\sin 2 \theta = 0.7\) for values of \(\theta\) between 0 and \(2 \pi\), giving your answers in radians correct to 3 significant figures.

6 Solve the equation \(\tan \theta = 2 \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

7 Showing your method clearly, solve the equation \(4 \sin ^ { 2 } \theta = 3 + \cos ^ { 2 } \theta\), for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

8 Show that the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) may be written in the form $$4 \sin ^ { 2 } \theta - \sin \theta = 0$$ Hence solve the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

9 Showing your method, solve the equation \(2 \sin ^ { 2 } \theta = \cos \theta + 2\) for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

10

1. Show that the equation \(2 \cos ^ { 2 } \theta + 7 \sin \theta = 5\) may be written in the form $$2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$$
2. By factorising this quadratic equation, solve the equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\). [4]

2. Solve the equation $$3 \operatorname { cosec } \theta ^ { \circ } + 8 \cos \theta ^ { \circ } = 0$$ for \(\theta\) in the interval \(0 \leq \theta \leq 180\), giving your answers to 1 decimal place.

6. (i) Express \(\sqrt { 3 } \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(ii) State the maximum value of \(\sqrt { 3 } \sin \theta + \cos \theta\) and the smallest positive value of \(\theta\) for which this maximum value occurs.
(iii) Solve the equation $$\sqrt { 3 } \sin \theta + \cos \theta + \sqrt { 3 } = 0$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\), giving your answers in terms of \(\pi\).

2

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

3 Show that the equation \(4 \cos ^ { 2 } \theta = 1 + \sin \theta\) can be expressed as $$4 \sin ^ { 2 } \theta + \sin \theta - 3 = 0$$ Hence solve the equation for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$

6 Solve the equation \(\sin 2 x = - 0.5\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

10

1. Sketch the graph of \(y = \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   On the same axes, sketch the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). Label each graph clearly.
2. Solve the equation \(\cos 2 x = 0.5\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

12

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. Giving your answers to 1 decimal place, solve the equation

$$5 \tan ^ { 2 } 2 \theta - 13 \sec 2 \theta = 1 ,$$ for \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).

4. Find the values of \(x\) in the interval \(- 180 < x < 180\) for which $$\tan ( x + 45 ) ^ { \circ } - \tan x ^ { \circ } = 4 ,$$ giving your answers to 1 decimal place.

2 Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(\sec ^ { 2 } \theta = 4 \tan \theta - 2\).

5

1. Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(4 \cos \theta - \sin \theta = 2\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).

3

1. Solve, for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\), the equation \(\sec \frac { 1 } { 2 } \alpha = 4\).
2. Solve, for \(0 ^ { \circ } < \beta < 180 ^ { \circ }\), the equation \(\tan \beta = 7 \cot \beta\).

9

1. Use the identity for \(\cos ( A + B )\) to prove that $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) \equiv \sqrt { 3 } - 2 \sin 2 \theta .$$
2. Hence find the exact value of \(4 \cos 82.5 ^ { \circ } \cos 52.5 ^ { \circ }\).
3. Solve, for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation \(4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = 1\).
4. Given that there are no values of \(\theta\) which satisfy the equation $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = k ,$$ determine the set of values of the constant \(k\).

5

1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac { 7 } { 2 }\), giving all solutions for which \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

7

1. Write down the formula for \(\cos 2 x\) in terms of \(\cos x\).
2. Prove the identity \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } \equiv 4 - 2 \sec ^ { 2 } x\).
3. Solve, for \(0 < x < 2 \pi\), the equation \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } = 3 \tan x - 7\).