1.05l Double angle formulae: and compound angle formulae

575 questions

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Edexcel C3 Q4
7 marks Moderate -0.3
  1. Prove, by counter-example, that the statement "\(\sec(A + B) = \sec A + \sec B\), for all \(A\) and \(B\)" is false. [2]
  2. Prove that $$\tan \theta + \cot \theta = 2 \cosec 2\theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [5]
Edexcel C3 Q8
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta, \quad \theta \neq \frac{n\pi}{2}, \quad n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
Edexcel C3 Q16
8 marks Standard +0.3
  1. Express \(1.5 \sin 2x + 2 \cos 2x\) in the form \(R \sin (2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate. [4]
  2. Express \(3 \sin x \cos x + 4 \cos^2 x\) in the form \(a \cos 2x + b \sin 2x + c\), where \(a\), \(b\) and \(c\) are constants to be found. [2]
  3. Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos^2 x\). [2]
Edexcel C3 Q23
8 marks Standard +0.3
  1. Express \(\sin x + \sqrt{3} \cos x\) in the form \(R \sin (x + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
  2. Show that the equation \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin x + \sqrt{3} \cos x = 2 \sin 2x.$$ [3]
  3. Deduce from parts (a) and (b) that \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin 2x - \sin (x + 60°) = 0.$$ [1]
Edexcel C3 Q30
4 marks Moderate -0.3
Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta.$$ [4]
Edexcel C4 Q18
7 marks Moderate -0.3
  1. Use integration by parts to find $$\int x \cos 2x \, dx.$$ [4]
  2. Prove that the answer to part \((a)\) may be expressed as $$\frac{1}{2} \sin x (2x \cos x - \sin x) + C,$$ where \(C\) is an arbitrary constant. [3]
Edexcel FP3 2014 June Q6
10 marks Challenging +1.3
[In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.] The points \(P(3\cos \alpha, 2\sin \alpha)\) and \(Q(3\cos \beta, 2\sin \beta)\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$
  1. Show the equation of the chord \(PQ\) is $$\frac{x}{3}\cos\frac{(\alpha + \beta)}{2} + \frac{y}{2}\sin\frac{(\alpha + \beta)}{2} = \cos\frac{(\alpha - \beta)}{2}$$ [4]
  2. Write down the coordinates of the mid-point of \(PQ\). [1]
Given that the gradient, \(m\), of the chord \(PQ\) is a constant,
  1. show that the centre of the chord lies on a line $$y = -kx$$ expressing \(k\) in terms of \(m\). [5]
Edexcel M1 Q2
8 marks Standard +0.3
\includegraphics{figure_1} A particle has mass \(2\) kg. It is attached at \(B\) to the ends of two light inextensible strings \(AB\) and \(BC\). When the particle hangs in equilibrium, \(AB\) makes an angle of \(30°\) with the vertical, as shown in Fig. 1. The magnitude of the tension in \(BC\) is twice the magnitude of the tension in \(AB\).
  1. Find, in degrees to one decimal place, the size of the angle that \(BC\) makes with the vertical. [4]
  2. Hence find, to 3 significant figures, the magnitude of the tension in \(AB\). [4]
Edexcel C2 Q4
9 marks Moderate -0.3
  1. Write down formulae for sin (A + B) and sin (A - B). Using X = A + B and Y = A - B, prove that $$\sin X + \sin Y = 2 \sin \frac{X + Y}{2} \cos \frac{X - Y}{2}.$$ [4 marks]
  2. Hence, or otherwise, solve, for 0 ≤ θ < 360, $$\sin 40° + \sin 20° = 0.$$ [5 marks]
OCR C2 Q9
12 marks Standard +0.2
    1. Write down the exact values of \(\cos \frac{1}{6}\pi\) and \(\tan \frac{1}{6}\pi\) (where the angles are in radians). Hence verify that \(x = \frac{1}{6}\pi\) is a solution of the equation $$2 \cos x = \tan 2x.$$ [3]
    2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2x.$$ [4]
    1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.) [4]
    2. State with a reason whether this approximation is an underestimate or an overestimate. [1]
OCR MEI C2 2014 June Q9
3 marks Moderate -0.3
Solve the equation \(\tan 2\theta = 3\) for \(0° < \theta < 360°\). [3]
Edexcel C2 Q8
10 marks Standard +0.3
  1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
  2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos(2x - \frac{\pi}{6}) = \frac{1}{2}.$$ [7]
Edexcel C3 Q4
6 marks Moderate -0.3
Prove that $$\frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos 2\theta$$ [6]
Edexcel C3 Q7
8 marks Standard +0.3
  1. Express \(\sin x + \sqrt{3} \cos x\) in the form \(R \sin (x + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
  2. Show that the equation \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin x + \sqrt{3} \cos x = 2 \sin 2x.$$ [3]
  3. Deduce from parts (a) and (b) that \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin 2x - \sin (x + 60°) = 0.$$ [1]
Edexcel C3 Q5
8 marks Moderate -0.3
  1. Given that \(\sin x = \frac{3}{5}\), use an appropriate double angle formula to find the exact value of \(\sec 2x\). [4]
  2. Prove that $$\cot 2x + \cosec 2x \equiv \cot x, \quad \left(x \neq \frac{n\pi}{2}, n \in \mathbb{Z}\right).$$ [4]
Edexcel C3 Q5
7 marks Standard +0.3
  1. Prove, by counter-example, that the statement "\(\sec(A + B) \equiv \sec A + \sec B\), for all \(A\) and \(B\)" is false [2]
  2. Prove that $$\tan \theta + \cot \theta = 2\cosec 2\theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [5]
Edexcel C3 Q6
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} \equiv \tan \theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
OCR C3 Q7
9 marks Standard +0.3
  1. Write down the formula for \(\cos 2x\) in terms of \(\cos x\). [1]
  2. Prove the identity \(\frac{4 \cos 2x}{1 + \cos 2x} = 4 - 2 \sec^2 x\). [3]
  3. Solve, for \(0 < x < 2\pi\), the equation \(\frac{4 \cos 2x}{1 + \cos 2x} = 3 \tan x - 7\). [5]
OCR C3 Q9
13 marks Challenging +1.2
  1. By first writing \(\sin 3\theta\) as \(\sin(2\theta + \theta)\), show that $$\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta.$$ [4]
  2. Determine the greatest possible value of $$9 \sin(\frac{10}{3}\alpha) - 12 \sin^3(\frac{10}{3}\alpha),$$ and find the smallest positive value of \(\alpha\) (in degrees) for which that greatest value occurs. [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(3 \sin 6\beta \cos 2\beta = 4\). [6]
OCR C3 Q5
7 marks Moderate -0.3
  1. Write down the identity expressing \(\sin 2\theta\) in terms of \(\sin \theta\) and \(\cos \theta\). [1]
  2. Given that \(\sin \alpha = \frac{1}{4}\) and \(\alpha\) is acute, show that \(\sin 2\alpha = \frac{1}{8}\sqrt{15}\). [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(5 \sin 2\beta \sec \beta = 3\). [3]
OCR C3 Q2
5 marks Moderate -0.3
It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac{12}{13}\). Find the exact value of
  1. \(\cot \theta\), [2]
  2. \(\cos 2\theta\). [3]
OCR C3 Q9
12 marks Standard +0.8
  1. Prove the identity $$\tan(\theta + 60°) \tan(\theta - 60°) \equiv \frac{\tan^2 \theta - 3}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Solve, for \(0° < \theta < 180°\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = 4 \sec^2 \theta - 3,$$ giving your answers correct to the nearest \(0.1°\). [5]
  3. Show that, for all values of the constant \(k\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = k^2$$ has two roots in the interval \(0° < \theta < 180°\). [3]
OCR C3 2010 January Q2
8 marks Standard +0.3
The angle \(\theta\) is such that \(0° < \theta < 90°\).
  1. Given that \(\theta\) satisfies the equation \(6 \sin 2\theta = 5 \cos \theta\), find the exact value of \(\sin \theta\). [3]
  2. Given instead that \(\theta\) satisfies the equation \(8 \cos \theta \cosec^2 \theta = 3\), find the exact value of \(\cos \theta\). [5]
OCR C3 2010 January Q9
12 marks Challenging +1.2
The value of \(\tan 10°\) is denoted by \(p\). Find, in terms of \(p\), the value of
  1. \(\tan 55°\), [3]
  2. \(\tan 5°\), [4]
  3. \(\tan \theta\), where \(\theta\) satisfies the equation \(3 \sin(\theta + 10°) = 7 \cos(\theta - 10°)\). [5]
OCR C3 2013 January Q2
5 marks Moderate -0.3
The acute angle \(A\) is such that \(\tan A = 2\).
  1. Find the exact value of \(\cosec A\). [2]
  2. The angle \(B\) is such that \(\tan (A + B) = 3\). Using an appropriate identity, find the exact value of \(\tan B\). [3]