1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1

Jia has to solve the equation $$2 - 2\sin^2 \theta = \cos \theta$$ where \(-180° \leq \theta \leq 180°\) Jia's working is as follows: $$2 - 2(1 - \cos^2 \theta) = \cos \theta$$ $$2 - 2 + 2\cos^2 \theta = \cos \theta$$ $$2\cos^2 \theta = \cos \theta$$ $$2\cos \theta = 1$$ $$\cos \theta = 0.5$$ $$\theta = 60°$$ Jia's teacher tells her that her solution is incomplete.

1. Explain the two errors that Jia has made. [2 marks]
2. Write down all the values of \(\theta\) that satisfy the equation $$2 - 2\sin^2 \theta = \cos \theta$$ where \(-180° \leq \theta \leq 180°\) [2 marks]

Find all the solutions of the equation $$\cos^2 \theta = 10 \sin \theta + 4$$ for \(0° < \theta < 360°\), giving your answers to the nearest degree. Fully justify your answer. [5 marks]

It is given that \(5\cos^2 \theta - 4\sin^2 \theta = 0\)

1. Find the possible values of \(\tan \theta\), giving your answers in exact form. [3 marks]
2. Hence, or otherwise, solve the equation $$5\cos^2 \theta - 4\sin^2 \theta = 0$$ giving all solutions of \(\theta\) to the nearest \(0.1°\) in the interval \(0° \leq \theta \leq 360°\) [2 marks]

Jessica, a maths student, is asked by her teacher to solve the equation \(\tan x = \sin x\), giving all solutions in the range \(0° \leq x \leq 360°\) The steps of Jessica's working are shown below. \(\tan x = \sin x\) Step 1 \(\Rightarrow\) \(\frac{\sin x}{\cos x} = \sin x\) Write \(\tan x\) as \(\frac{\sin x}{\cos x}\) Step 2 \(\Rightarrow\) \(\sin x = \sin x \cos x\) Multiply by \(\cos x\) Step 3 \(\Rightarrow\) \(1 = \cos x\) Cancel \(\sin x\) \(\Rightarrow\) \(x = 0°\) or \(360°\) The teacher tells Jessica that she has not found all the solutions because of a mistake. Explain why Jessica's method is not correct. [2 marks]

It is given that \(\cos 15° = \frac{1}{2}\sqrt{2 + \sqrt{3}}\) and \(\sin 15° = \frac{1}{2}\sqrt{2 - \sqrt{3}}\) Show that \(\tan^2 15°\) can be written in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. Fully justify your answer. [3 marks]

Find all the solutions of $$9 \sin^2 x - 6 \sin x + \cos^2 x = 0$$ where \(0° \leq x \leq 180°\) Give your solutions to the nearest degree. Fully justify your answer. [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 \(\frac{3}{4}\)

It is given that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\) and \(\cos 15° = \frac{\sqrt{6} + \sqrt{2}}{4}\) Use these two expressions to show that \(\tan 15° = 2 - \sqrt{3}\) Fully justify your answer. [4 marks]

One of the equations below is true for all values of \(x\) Identify the correct equation. Tick (\(\checkmark\)) one box. [1 mark] \(\cos^2 x = -1 - \sin^2 x\) \(\square\) \(\cos^2 x = -1 + \sin^2 x\) \(\square\) \(\cos^2 x = 1 - \sin^2 x\) \(\square\) \(\cos^2 x = 1 + \sin^2 x\) \(\square\)

Solve the equation $$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0° < \theta < 180°\) Fully justify your answer. [5 marks]

Prove the identity \(\cot^2 \theta - \cos^2 \theta = \cot^2 \theta \cos^2 \theta\) [3 marks]

It is given that $$(2 \sin \theta + 3 \cos \theta)^2 + (6 \sin \theta - \cos \theta)^2 = 30$$ and that \(\theta\) is obtuse. Find the exact value of \(\sin \theta\). Fully justify your answer. [6 marks]

A wooden frame is to be made to support some garden decking. The frame is to be in the shape of a sector of a circle. The sector \(OAB\) is shown in the diagram, with a wooden plank \(AC\) added to the frame for strength. \(OA\) makes an angle of \(\theta\) with \(OB\). \includegraphics{figure_2}

1. Show that the exact value of \(\sin\theta\) is \(\frac{4\sqrt{14}}{15}\) [3 marks]
2. Write down the value of \(\theta\) in radians to 3 significant figures. [1 mark]
3. Find the area of the garden that will be covered by the decking. [2 marks]

Solve, for \(360° \leqslant x < 540°\), $$12\sin^2 x + 7\cos x - 13 = 0$$ Give your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]

In a triangle \(PQR\), \(PQ = 20\) cm, \(PR = 10\) cm and angle \(QPR = \theta\), where \(\theta\) is measured in degrees. The area of triangle \(PQR\) is 80 cm\(^2\).

1. Show that the two possible values of \(\cos \theta = \pm \frac{3}{5}\) [4]

Given that \(QR\) is the longest side of the triangle,

1. find the exact perimeter of the triangle \(PQR\), giving your answer as a simplified surd. [3]

The complex numbers \(z\) and \(w\) are defined by $$z = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4}$$ and $$w = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6}$$ By evaluating the product \(zw\), show that $$\tan\frac{5\pi}{12} = 2 + \sqrt{3}$$ [6 marks]

Solve the following equation for values of \(\theta\) between \(0°\) and \(360°\). $$3\sin^2 \theta - 5\cos^2 \theta = 2\cos \theta$$ [7]