1.05l Double angle formulae: and compound angle formulae

Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for

1. \(\cos \theta\), [1]
2. \(\tan \theta\), [2]
3. \(\sin(\theta + \pi)\). [1]

Solve the equation \(\tan(\theta - 60°) = 3 \cot \theta\) for \(-90° < \theta < 90°\). [5]

It is given that \(\theta\) is an acute angle measured in degrees such that $$2\sec^2\theta + 3\tan\theta = 22.$$

1. Find the value of \(\tan\theta\). [3]
2. Use an appropriate formula to find the exact value of \(\tan(\theta + 135°)\). [3]

\includegraphics{figure_6} The diagram shows part of the curve with equation $$y = 4\sin^2 x + 8\sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\). [3]
2. Show that the equation of the curve can be written $$y = 5 + 8\sin x - 2\cos 2x,$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]

A curve has equation \(y = 2\sin 2x - 5\cos 2x + 6\) and is defined for \(0 \leq x \leq \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. [6]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sin 2x + 3\cos 2x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3.

1. Find an expression for \(\frac{dy}{dx}\). [2]
2. By first expressing \(\frac{dy}{dx}\) in the form \(R\cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 significant figures. [8]

By first expressing the equation \(\tan(x + 45°) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0° < x < 180°\). [6]

Solve the equation $$\tan(45° - x) = 2\tan x,$$ giving all solutions in the interval \(0° < x < 180°\). [5]

Solve the equation $$\cos(x + 30°) = 2\cos x,$$ giving all solutions in the interval \(-180° < x < 180°\). [5]

\includegraphics{figure_6} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(OAB\) is \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.

1. Show that \(\cos 2\theta = \frac{2\sin 2\theta - \pi}{4\theta}\). [5]
2. Use the iterative formula $$\theta_{n+1} = \frac{1}{2}\cos^{-1}\left(\frac{2\sin 2\theta_n - \pi}{4\theta_n}\right),$$ with initial value \(\theta_1 = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places. [3]

Showing all necessary working, solve the equation \(\sin(\theta - 30°) + \cos \theta = 2 \sin \theta\), for \(0° < \theta < 180°\). [4]

Given that \(z = e^{i\theta}\) and \(n\) is a positive integer, show that $$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2] Hence express \(\sin^6 \theta\) in the form $$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$ where the constants \(p\), \(q\), \(r\), \(s\) are to be determined. [4] Hence find the mean value of \(\sin^6 \theta\) with respect to \(\theta\) over the interval \(0 \leq \theta \leq \frac{1}{4}\pi\). [5]

The curve \(C\) has polar equation \(r = a \cos 3\theta\), for \(-\frac{1}{6}\pi \leqslant \theta \leqslant \frac{1}{6}\pi\), where \(a\) is a positive constant.

1. Sketch \(C\). [2]
2. Find the area of the region enclosed by \(C\), showing full working. [3]
3. Using the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\), find a cartesian equation of \(C\). [3]

Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which

1. \(\cos (\theta + 75)° = 0\). [3]
2. \(\sin 2\theta ° = 0.7\), giving your answers to one decimal place. [5]