1.05a Sine, cosine, tangent: definitions for all arguments

A parallelogram has sides of length 6 cm and 4.5 cm. The larger interior angles of the parallelogram have size \(\alpha\) Given that the area of the parallelogram is 24 cm², find the exact value of \(\tan \alpha\) [4 marks]

It is given that \(\sin \theta = \frac{4}{5}\) and \(90° < \theta < 180°\) Find the value of \(\cos \theta\) Circle your answer. [1 mark] \(-\frac{3}{4}\) \qquad \(-\frac{3}{5}\) \qquad \(\frac{3}{5}\) \qquad \(\frac{3}{4}\)

In this question take \(g = 10\). A small ball P is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation of \((\alpha + \theta)\) from a point O at the bottom of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\) and \(\tan \theta = \frac{3}{4}\). The ball subsequently hits the plane at a point A, where OA is a line of greatest slope of the plane, as shown in the diagram. \includegraphics{figure_9}

1. Determine the following, in either order.
   [9]

After P hits the plane at A it continues to move away from O. Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.

1. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree. [3]

Find a small positive value of \(x\) which is an approximate solution of the equation. $$\cos x - 4\sin x = x^2.$$ [4]

The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_2}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
   2. Find the value of \(r\) at the point \(P\). [1]
   3. Mark the point \(P\) on a copy of the graph. [1]
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]

Show that $$\sum_{n=2}^{\infty} \left(\frac{1}{4}\right)^n \cos(180n)^{\circ} = \frac{9}{28}$$ [3]