1.05o Trigonometric equations: solve in given intervals

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 \(0 \leqslant \theta \leqslant 2 \pi\).

5

1. Show that the equation \(4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x\) can be expressed in the form $$3 \sin ^ { 2 } x - 4 \sin x + 1 = 0$$
2. Hence find all values of \(x\) for which \(0 < x < \pi\) that satisfy the equation $$4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x$$

4

1. Sketch the graph of \(y = \sqrt { 2 } \sin x\) for \(0 \leqslant x \leqslant 2 \pi\). The points \(P\) and \(Q\) on the graph have \(x\)-coordinates \(\frac { 1 } { 4 } \pi\) and \(\frac { 3 } { 4 } \pi\), respectively.
2. Determine the equation of the tangent to the curve at \(P\). The normals to the curve at \(P\) and \(Q\) intersect at the point \(R\).
3. Determine the exact coordinates of \(R\).

Solve the following equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$ a) Given that \(y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\). b) Find \(\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x\). The diagram below shows a sketch of \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of \(y = 4 + f ( x )\), clearly indicating any asymptotes.
b) Sketch the graph of \(y = f ( x - 3 )\), clearly indicating any asymptotes.
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0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve \(C\) with equation \(y = 14 + 5 x - x ^ { 2 }\) and line \(L\) with equation \(y = x + 2\). The line intersects the curve at the points \(A\) and \(B\).
a) Find the coordinates of \(A\) and \(B\).
b) Calculate the area enclosed by \(L\) and \(C\). Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$

a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that $$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$ b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation $$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$

Solve the equation $$6 \sec ^ { 2 } x - 8 = \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

Find the exact solution of the equation $$\cos\frac{x}{6} + \tan 2x + \frac{\sqrt{3}}{2} = 0 \text{ for } -\frac{1}{4}\pi < x < \frac{1}{4}\pi.$$ [2]

Solve the equation \(4\sin^4\theta + 12\sin^2\theta - 7 = 0\) for \(0° \leqslant \theta \leqslant 360°\). [4]

The function \(f : x \mapsto 4 - 3\sin x\) is defined for the domain \(0 \leq x < 2\pi\).

1. Solve the equation \(f(x) = 2\). [3]
2. Sketch the graph of \(y = f(x)\). [2]
3. Find the set of values of \(k\) for which the equation \(f(x) = k\) has no solution. [2]

The function \(g : x \mapsto 4 - 3\sin x\) is defined for the domain \(\frac{1}{2}\pi \leq x \leq A\).

1. State the largest value of \(A\) for which \(g\) has an inverse. [1]
2. For this value of \(A\), find the value of \(g^{-1}(3)\). [2]

The function \(f\) is such that \(f(x) = 3 - 4\cos^k x\), for \(0 \leq x \leq \pi\), where \(k\) is a constant.

1. In the case where \(k = 2\),
   1. find the range of \(f\), [2]
   2. find the exact solutions of the equation \(f(x) = 1\). [3]
2. In the case where \(k = 1\),
   1. sketch the graph of \(y = f(x)\), [2]
   2. state, with a reason, whether \(f\) has an inverse. [1]

The function \(\text{f} : x \mapsto 5 + 3\cos(\frac{1}{3}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).

1. Solve the equation \(\text{f}(x) = 7\), giving your answer correct to 2 decimal places. [3]
2. Sketch the graph of \(y = \text{f}(x)\). [2]
3. Explain why \(\text{f}\) has an inverse. [1]
4. Obtain an expression for \(\text{f}^{-1}(x)\). [3]

A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, \(h\) m, of a passenger above the ground is given by the formula \(h = 60(1 - \cos kt)\). In this formula, \(k\) is a constant, \(t\) is the time in minutes that has elapsed since the passenger started the ride at ground level and \(kt\) is measured in radians.

1. Find the greatest height of the passenger above the ground. [1]

One complete revolution of the wheel takes 30 minutes.

1. Show that \(k = \frac{\pi}{15}\pi\). [2]
2. Find the time for which the passenger is above a height of 90 m. [3]

\includegraphics{figure_9} The function f : \(x \mapsto p \sin^2 2x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \text{f}(x)\).

1. In terms of \(p\) and \(q\), state the range of f. [2]
2. State the number of solutions of the following equations.
   1. \(\text{f}(x) = p + q\) [1]
   2. \(\text{f}(x) = q\) [1]
   3. \(\text{f}(x) = \frac{1}{2}p + q\) [1]
3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\text{f}(x) = 4\), showing all necessary working. [5]