Trigonometric equations in context

5. Solve, for \(0 < \theta < 360 ^ { \circ }\),
a) \(5 \cos ( \theta + 30 ) = 3\) b) \(\cos ^ { 2 } ( x ) + 4 \sin ^ { 2 } ( x ) + 4 \sin ( x ) = 0\) Give each non-exact solution to one decimal place.
[0pt]

1. (a) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which

$$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3$$ (b) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$\tan ^ { 2 } \theta = 4$$

12. [In this question solutions based entirely on graphical or numerical methods are not acceptable.]

1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin \left( x + 65 ^ { \circ } \right) + 2 = 0$$ giving your answers in degrees to one decimal place.
2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$12 \sin ^ { 2 } \theta + \cos \theta = 6$$ giving your answers in radians to 3 significant figures.

4 The equation of a curve is \(y = 2 x - \tan x\), where \(x\) is in radians. Find the coordinates of the stationary points of the curve for which \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).

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 \(0 ^ { \circ } < x < 180 ^ { \circ }\).

11.

1. Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$ (4)
2. Hence or otherwise solve, for \(0 \leqslant \theta < 180\), $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.
   (3)

2 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. \(\sin \frac { 1 } { 2 } x = 0.8\)
2. \(\sin x = 3 \cos x\)

4

1. On the axes in the Printed Answer Booklet, sketch the graph of \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve the equation \(\sin 2 \theta = - \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\). \(5 M\) is the event that an A-level student selected at random studies mathematics. \(C\) is the event that an A-level student selected at random studies chemistry.
   You are given that \(\mathrm { P } ( M ) = 0.42 , \mathrm { P } ( C ) = 0.36\) and \(\mathrm { P } ( \mathrm { M }\) and \(\mathrm { C } ) = 0.24\). These probabilities are shown in the two-way table below.

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(M\)	\(M ^ { \prime }\)	Total
   \(C\)	0.24		0.36
   \(C ^ { \prime }\)
   Total	0.42		1

10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
4. State the largest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

5 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
2. \(\sin 2 x = - \cos 2 x\).

11. In this question solutions based entirely on graphical or numerical methods are not acceptable.

1. Solve, for \(0 \leqslant x < 2 \pi\), $$3 \cos ^ { 2 } x + 1 = 4 \sin ^ { 2 } x$$ giving your answers in radians to 2 decimal places.
2. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$5 \sin \left( \theta + 10 ^ { \circ } \right) = \cos \left( \theta + 10 ^ { \circ } \right)$$ giving your answers in degrees to one decimal place.

7

1. Solve the equation \(2 \cos ^ { 2 } \theta = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
2. The smallest positive solution of the equation \(2 \cos ^ { 2 } ( n \theta ) = 3 \sin ( n \theta )\), where \(n\) is a positive integer, is \(10 ^ { \circ }\). State the value of \(n\) and hence find the largest solution of this equation in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

7

1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$
2. Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.

6 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

9. (a) Sketch, for \(0 \leqslant x \leqslant 2 \pi\), the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.
(c) Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 6 } \right) = 0.65$$ giving your answers in radians to 2 decimal places.

8. (i) Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
(3 marks)
(ii) Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
(2 marks)

7. (i) Solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), $$( 1 + \tan \theta ) ( 5 \sin \theta - 2 ) = 0$$ (ii) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \sin x = 3 \tan x .$$

5. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 10 + 5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.

1. Show that the height of the water 1 hour after midnight is 12.5 metres.
2. Find, to the nearest minute, the times before midday when the height of the water is 9 metres.

3 Simplify \(\frac { \sqrt { 1 - \cos ^ { 2 } \theta } } { \tan \theta }\), where \(\theta\) is an acute angle.
[0pt] [3]

3 In this question you must show detailed reasoning.

1. Solve the equation \(4 \sin ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
2. Prove that \(\frac { \sin ^ { 2 } \theta - 1 + \cos \theta } { 1 - \cos \theta } \equiv \cos \theta \quad ( \cos \theta \neq 1 )\).

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

1 Use an isosceles right-angled triangle to show that \(\cos 45 ^ { \circ } = \frac { 1 } { \sqrt { 2 } }\).

4 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ }$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).

1. (i) Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation,

$$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
(ii) (a) A student's attempt at the question
"Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x \\ 7 \times \frac { \sin x } { \cos x } = 8 \sin x \\ 7 \sin x = 8 \sin x \cos x \\ 7 = 8 \cos x \\ \cos x = \frac { 7 } { 8 } \\ x = 29.0 ^ { \circ } \text { (to } 3 \text { sf) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$

4

1. Solve the equation \(4 \sin ^ { 2 } x + 8 \cos x - 7 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Hence find the solution of the equation \(4 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) + 8 \cos \left( \frac { 1 } { 2 } \theta \right) - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3 Given that \(\sin \theta = \frac { \sqrt { 3 } } { 4 }\), find in surd form the possible values of \(\cos \theta\).

5 The function f is such that \(\mathrm { f } ( x ) = 2 \sin ^ { 2 } x - 3 \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\).

1. Express \(\mathrm { f } ( x )\) in the form \(a + b \cos ^ { 2 } x\), stating the values of \(a\) and \(b\).
2. State the greatest and least values of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) + 1 = 0\).

1 Express \(6 \cos 2 \theta + \sin \theta\) in terms of \(\sin \theta\).
Hence solve the equation \(6 \cos 2 \theta + \sin \theta = 0\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

9

1. \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_362_979_1505_625} The diagram shows part of the curve \(y = \cos 2 x\), where \(x\) is in radians. The point \(A\) is the minimum point of this part of the curve.
   1. State the period of \(y = \cos 2 x\).
   2. State the coordinates of \(A\).
   3. Solve the inequality \(\cos 2 x \leqslant 0.5\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.
2. Solve the equation \(\cos 2 x = \sqrt { 3 } \sin 2 x\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.