1.05o Trigonometric equations: solve in given intervals

5 Find the general solutions of the following equations, giving your answers in terms of \(\pi\) :

1. \(\quad \tan 3 x = \sqrt { 3 }\);
2. \(\quad \tan \left( 3 x - \frac { \pi } { 3 } \right) = - \sqrt { 3 }\).

4 Find, in radians, the general solution of the equation $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in terms of \(\pi\).

2.Solve,for \(0 \leqslant x \leqslant 360 ^ { \circ }\) $$\sin 47 ^ { \circ } \cos ^ { 3 } x + \cos 47 ^ { \circ } \sin x \cos ^ { 2 } x = \frac { 1 } { 2 } \cos ^ { 2 } x$$

8

1. Show that \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = \sin 2 \theta\).
2. In this question you must show detailed reasoning. Solve \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = 3 \cos 2 \theta\) for \(0 \leq \theta \leq \pi\).

3 In this question you must show detailed reasoning. Given that \(5 \sin 2 x = 3 \cos x\), where \(0 ^ { \circ } < x < 90 ^ { \circ }\), find the exact value of \(\sin x\).

2 Given that $$\cos \left( \theta - 20 ^ { \circ } \right) = \cos 60 ^ { \circ }$$ which one of the following is a possible value for \(\theta\) ?
Circle your answer.
[0pt] [1 mark] \(40 ^ { \circ }\) \(140 ^ { \circ }\) \(280 ^ { \circ }\) \(320 ^ { \circ }\)

8

1. Given that $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ show that $$8 \cot ^ { 2 } \theta - 2 \cot \theta - 1 = 0$$ 8
2. Hence, solve $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ in the interval \(0 < \theta < 2 \pi\) Give your answers to two decimal places.
   8
3. Solve $$9 \sin ^ { 2 } \left( 2 x - \frac { \pi } { 4 } \right) + \sin \left( 4 x - \frac { \pi } { 2 } \right) = 8$$ in the interval \(0 < x < \frac { \pi } { 2 }\) Give your answers to one decimal place.

10

1. Point \(A\) on the curve has coordinates ( \(a , 0.5\) )
   10
   1. (i) Find the value of \(a\) 10
   2. (ii) State the value of \(\sin \left( 180 ^ { \circ } - a ^ { \circ } \right)\) 10
   3. Point \(B\) on the curve has coordinates \(\left( b , - \frac { 3 } { 7 } \right)\) 10
   4. 1. Find the exact value of \(\sin \left( b ^ { \circ } - 180 ^ { \circ } \right)\) 10
   5. (ii) Find the exact value of \(\cos b ^ { \circ }\)

13

1. Two of the solutions to the equation \(\cos 6 \theta = 0\) are \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) Find the other solutions to the equation \(\cos 6 \theta = 0\) for \(0 \leq \theta \leq \pi\) 13
2. Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ 13
3. Use the fact that \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) are solutions to the equation \(\cos 6 \theta = 0\) to find a factor of \(32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1\) in the form ( \(a \cos ^ { 2 } \theta + b\) ), where \(a\) and \(b\) are integers.
   [0pt] [4 marks]
4. Hence show that $$\cos \left( \frac { 11 \pi } { 12 } \right) = - \sqrt { \frac { 2 + \sqrt { 3 } } { 4 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-25_2492_1721_217_150}

1. (a) Show that the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) transforms the integral

$$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$ into the integral $$\int \frac { 1 } { 3 t ^ { 2 } + 2 t + 2 } \mathrm {~d} t$$ (b) Hence determine $$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$2 \tan \theta = 3 \cos \theta$$ can be written as $$3 \sin ^ { 2 } \theta + 2 \sin \theta - 3 = 0$$
2. Hence solve, for \(- \pi < x < \pi\), the equation $$2 \tan \left( 2 x + \frac { \pi } { 3 } \right) = 3 \cos \left( 2 x + \frac { \pi } { 3 } \right)$$ giving your answers to 3 significant figures.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), the equation $$3 \tan ^ { 2 } \theta + 7 \sec \theta - 3 = 0$$ giving your answers to one decimal place.

10

1. Prove that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$ and hence show that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
2. Hence solve the equation \(\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

8

1. Express \(\sin x - \sqrt { 8 } \cos x\) in the form \(R \sin ( x - \alpha )\) where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\).
2. Hence write down the maximum value of \(\sin x - \sqrt { 8 } \cos x\) and find the smallest positive value of \(x\) for which it occurs.

5 Solve \(\sin \left( 2 \theta + 30 ^ { \circ } \right) = 0.1\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

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 \(4 x ^ { 3 } - 2 x ^ { 2 } - 3 x + 1 = 0\).

11

1. Prove that $$\sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \frac { 1 } { 2 } \sin ^ { 2 } \theta - \frac { 3 } { 4 } = \frac { 1 } { 4 } \sqrt { 3 } \sin 2 \theta .$$
2. Hence solve the equation $$2 \sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \sin ^ { 2 } \theta = 1 \text { for } - \pi \leqslant \theta \leqslant \pi .$$

10

1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).

4 Solve the equation \(\sin 2 x = \sqrt { 3 } \cos x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3 Solve the equation \(\tan \left( \theta + 10 ^ { \circ } \right) = 0.1\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Sketch the graph of \(y = \sec \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve \(\sec \theta = \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).

6 \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-3_545_557_269_794} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) lies on \(O B\), and \(A C\) is perpendicular to \(O B\). The area of the triangle \(A O C\) is equal to the area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\).

1. Show that \(\theta = \sin \theta ( 1 + \cos \theta )\). The equation \(\theta = \sin \theta ( 1 + \cos \theta )\) has only one positive root.
2. Use an iterative process based on this equation to find the value of the root correct to 3 significant figures. Use a starting value of 1 and show the result of each iteration. Use a change of sign to verify that the value you have found is correct to 3 significant figures.

9 In this question, \(x\) denotes an angle measured in degrees.

1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).

10

1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
2. Hence solve the equation \(\cot \left( \theta + \frac { \neq } { 4 } \right) + \frac { \sin \left( \theta + \frac { \neq } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \neq } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).