Multiple angle equations

5

1. Given that \(4 \tan x + 3 \cos x + \frac { 1 } { \cos x } = 0\), show, without using a calculator, that \(\sin x = - \frac { 2 } { 3 }\).
2. Hence, showing all necessary working, solve the equation $$4 \tan \left( 2 x - 20 ^ { \circ } \right) + 3 \cos \left( 2 x - 20 ^ { \circ } \right) + \frac { 1 } { \cos \left( 2 x - 20 ^ { \circ } \right) } = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{78a9b100-c3bd-4054-b539-ec8304440063-10_551_641_260_751} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3 ,$$ where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(A\) and the shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\).

1. Find the exact \(x\)-coordinate of \(A\).
2. Find the exact gradient of the curve at \(A\).
3. Find the exact area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Solve the equation \(5 \sec ^ { 2 } 2 \theta = \tan 2 \theta + 9\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-26_643_736_721_660} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where \(A\) is a constant and \(\theta\) is measured in radians.
   The points \(P , Q\) and \(R\) lie on the curve and are shown in Figure 1.
   Given that the \(y\) coordinate of \(P\) is 7
   (a) state the value of \(A\),
   (b) find the exact coordinates of \(Q\),
   (c) find the value of \(\theta\) at \(R\), giving your answer to 3 significant figures.

5. Solve, for \(0 \leqslant \theta \leqslant 180 ^ { \circ }\), $$2 \cot ^ { 2 } 3 \theta = 7 \operatorname { cosec } 3 \theta - 5$$ Give your answers in degrees to 1 decimal place.

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 \(0 \leq \theta \leq 360 ^ { \circ }\).

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 \(\tan \beta = 7 \cot \beta\).

1 Solve the equation for values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). $$\cot 2 \theta = 5$$

2 Solve, correct to 2 decimal places, the equation \(\cot 2 \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

2. 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 \(0 \leq \theta \leq 360 ^ { \circ }\).