Solve tan·sin or tan·trig product

A question is this type if and only if it requires solving an equation involving a product such as tan θ · sin θ = k or sin x · tan x = k by rewriting using sin/cos and applying a Pythagorean identity to obtain a polynomial equation in cos.

4 questions · Standard +0.1

1.05o Trigonometric equations: solve in given intervals
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OCR C2 2012 January Q9
9 marks Standard +0.3
9
  1. Sketch the graph of \(y = \tan \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\) on the axes provided.
    On the same axes, sketch the graph of \(y = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\), indicating the point of intersection with the \(y\)-axis.
  2. Show that the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) can be expressed in the form $$3 \sin ^ { 2 } \left( \frac { 1 } { 2 } x \right) + \sin \left( \frac { 1 } { 2 } x \right) - 3 = 0$$ Hence solve the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).
Edexcel AS Paper 1 2020 June Q9
8 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-20_810_1214_255_427} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve with equation \(y = 3 \cos x ^ { \circ }\).
The point \(P ( c , d )\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.
  1. State the value of \(c\) and the value of \(d\).
  2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3 \cos x ^ { \circ }\) to the curve with equation
    1. \(y = 3 \cos \left( \frac { x ^ { \circ } } { 4 } \right)\)
    2. \(y = 3 \cos ( x - 36 ) ^ { \circ }\)
  3. Solve, for \(450 ^ { \circ } \leqslant \theta < 720 ^ { \circ }\), $$3 \cos \theta = 8 \tan \theta$$ giving your solution to one decimal place.
    In part (c) you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
AQA C2 2005 June Q8
10 marks Moderate -0.3
8
    1. Show that the equation $$4 \tan \theta \sin \theta = 15$$ can be written as $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ (1 mark)
    2. Use an appropriate identity to show that the equation $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ can be written as $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$
    1. Solve the equation \(4 c ^ { 2 } + 15 c - 4 = 0\).
    2. Hence explain why the only value of \(\cos \theta\) which satisfies the equation $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$ is \(\cos \theta = \frac { 1 } { 4 }\).
    3. Hence solve the equation \(4 \tan \theta \sin \theta = 15\) giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Write down all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\) for which $$4 \tan 4 x \sin 4 x = 15$$ giving your answers to the nearest degree.
AQA C2 2013 June Q9
14 marks Standard +0.3
9
    1. On the axes given below, sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    2. Solve the equation \(\tan x = - 1\), giving all values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    1. Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
    2. Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-5_720_1367_806_390}