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

710 questions

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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]
OCR C2 Q3
6 marks Moderate -0.8
  1. Given that $$5 \cos \theta - 2 \sin \theta = 0,$$ show that \(\tan \theta = 2.5\) [2]
  2. Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2x° - 2 \sin 2x° = 0,$$ giving your answers to 1 decimal place. [4]
OCR C2 Q5
8 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 $$\cos 3x = \frac{\sqrt{3}}{2}.$$ [5]
OCR MEI C2 Q4
3 marks Moderate -0.8
\(\theta\) is an acute angle and \(\sin \theta = \frac{1}{4}\). Find the exact value of \(\tan \theta\). [3]
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 Q7
12 marks Standard +0.3
    1. Express \((12 \cos \theta - 5 \sin \theta)\) in the form \(R \cos (\theta + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
  2. Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [3]
  3. Solve $$8 \cot \theta - 3 \tan \theta = 2,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [5]
Edexcel C3 Q1
4 marks Moderate -0.5
Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [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]
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 Q2
5 marks Standard +0.3
Solve, for \(0° < \theta < 360°\), the equation \(\sec^2 \theta = 4 \tan \theta - 2\). [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 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]
OCR C3 2013 January Q9
10 marks Standard +0.8
  1. Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
  2. Hence solve the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
  3. It is given that there are two values of \(\theta\), where \(-90° < \theta < 90°\), satisfying the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \frac{2}{3}\theta - \sin \frac{2}{3}\theta) = k,$$ where \(k\) is a constant. Find the set of possible values of \(k\). [3]
OCR C3 2009 June Q3
6 marks Standard +0.3
The angles \(\alpha\) and \(\beta\) are such that $$\tan \alpha = m + 2 \quad \text{and} \quad \tan \beta = m,$$ where \(m\) is a constant.
  1. Given that \(\sec^2 \alpha - \sec^2 \beta = 16\), find the value of \(m\). [3]
  2. Hence find the exact value of \(\tan(\alpha + \beta)\). [3]
OCR C3 2010 June Q3
6 marks Standard +0.3
  1. Express the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\) in the form \(a \sin^2 \theta + b \sin \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Hence solve, for \(-180° < \theta < 180°\), the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\). [3]
OCR MEI C3 2011 January Q4
3 marks Easy -1.2
Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}
OCR C3 Q8
10 marks Standard +0.8
  1. Sketch on the same diagram the graphs of $$y = \sin^{-1} x, \quad -1 \leq x \leq 1$$ and $$y = \cos^{-1} (2x), \quad -\frac{1}{2} \leq x \leq \frac{1}{2}.$$ [3]
Given that the graphs intersect at the point with coordinates \((a, b)\),
  1. show that \(\tan b = \frac{1}{2}\), [3]
  2. find the value of \(a\) in the form \(k\sqrt{5}\). [4]
OCR C3 Q3
6 marks Standard +0.8
Find all values of \(\theta\) in the interval \(-180 < \theta < 180\) for which $$\tan^2 \theta^\circ + \sec \theta^\circ = 1.$$ [6]
OCR MEI C3 Q6
18 marks Standard +0.3
The function \(\text{f}(x) = \frac{\sin x}{2 - \cos x}\) has domain \(-\pi \leqslant x \leqslant \pi\). Fig. 8 shows the graph of \(y = \text{f}(x)\) for \(0 \leqslant x \leqslant \pi\). \includegraphics{figure_6}
  1. Find \(\text{f}(-x)\) in terms of \(\text{f}(x)\). Hence sketch the graph of \(y = \text{f}(x)\) for the complete domain \(-\pi \leqslant x \leqslant \pi\). [3]
  2. Show that \(\text{f}'(x) = \frac{2\cos x - 1}{(2 - \cos x)^2}\). Hence find the exact coordinates of the turning point P. State the range of the function \(\text{f}(x)\), giving your answer exactly. [8]
  3. Using the substitution \(u = 2 - \cos x\) or otherwise, find the exact value of \(\int_0^\pi \frac{\sin x}{2 - \cos x} dx\). [4]
  4. Sketch the graph of \(y = \text{f}(2x)\). [1]
  5. Using your answers to parts (iii) and (iv), write down the exact value of \(\int_0^{\frac{\pi}{2}} \frac{\sin 2x}{2 - \cos 2x} dx\). [2]
OCR MEI C3 Q5
3 marks Easy -1.2
Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}
AQA C4 2010 June Q5
11 marks Standard +0.3
    1. Show that the equation \(3\cos 2x + 2\sin x + 1 = 0\) can be written in the form $$3\sin^2 x - \sin x - 2 = 0$$ [3 marks]
    2. Hence, given that \(3\cos 2x + 2\sin x + 1 = 0\), find the possible values of \(\sin x\). [2 marks]
    1. Express \(3\cos 2x + 2\sin 2x\) in the form \(R\cos(2x - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\), giving \(\alpha\) to the nearest \(0.1°\). [3 marks]
    2. Hence solve the equation $$3\cos 2x + 2\sin 2x + 1 = 0$$ for all solutions in the interval \(0° < x < 180°\), giving \(x\) to the nearest \(0.1°\). [3 marks]