1.05f Trigonometric function graphs: symmetries and periodicities

162 questions

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CAIE P3 2017 November Q4
7 marks Standard +0.3
  1. Prove the identity \(\tan(45° + x) + \tan(45° - x) = 2 \sec 2x\). [4]
  2. Hence sketch the graph of \(y = \tan(45° + x) + \tan(45° - x)\) for \(0° \leqslant x \leqslant 90°\). [3]
Edexcel C2 Q9
10 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 2\pi\), the graph of \(y = \sin\left(x + \frac{\pi}{6}\right)\). [2]
  2. Write down the exact coordinates of the points where the graph meets the coordinate axes. [3]
  3. Solve, for \(0 \leq x \leq 2\pi\), the equation \(\sin\left(x + \frac{\pi}{6}\right) = 0.65\), giving your answers in radians to 2 decimal places. [5]
Edexcel C2 Q24
10 marks Moderate -0.8
$$f(x) = 5 \sin 3x°, \quad 0 \leq x \leq 180.$$
  1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis. [3]
  2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
  3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]
Edexcel C2 Q36
8 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
  2. Write down the coordinates of the points at which the graph meets the axes. [3]
  3. Solve, for \(0 \leq x < 360°\), the equation $$\sin (x + 30°) = -\frac{1}{2}.$$ [3]
Edexcel C2 Q4
8 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
  2. Write down the coordinates of the points at which the graph meets the axes. [3]
  3. Solve, for \(0 \leq x < 360°\), the equation $$\sin (x + 30°) = -\frac{1}{2}.$$ [3]
Edexcel C2 Q7
9 marks Moderate -0.8
The curve C has equation \(y = \cos \left(x + \frac{\pi}{4}\right)\), \(0 \leq x \leq 2\pi\).
  1. Sketch C. [2]
  2. Write down the exact coordinates of the points at which C meets the coordinate axes. [3]
  1. Solve, for x in the interval \(0 \leq x \leq 2\pi\), $$\cos \left(x + \frac{\pi}{4}\right) = 0.5,$$ giving your answers in terms of π. [4]
Edexcel C2 Q7
10 marks Moderate -0.3
$$f(x) = 5\sin 3x°, \quad 0 \leq x \leq 180.$$
  1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis [3]
  2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
  3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]
Edexcel C2 Q4
8 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
  2. Write down the coordinates of the points at which the graph meets the axes. [3]
  3. Solve, for \(0 \leq x < 360°\), the equation \(\sin (x + 30°) = -\frac{1}{2}\). [3]
Edexcel C2 Q3
9 marks Moderate -0.8
The curve \(C\) has equation \(y = \cos \left( x + \frac{\pi}{4} \right)\), \(0 \leq x \leq 2\pi\).
  1. Sketch \(C\). [2]
  2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes. [3]
  3. Solve, for \(x\) in the interval \(0 \leq x \leq 2\pi\), \(\cos \left( x + \frac{\pi}{4} \right) = 0.5\), giving your answers in terms of \(\pi\). [4]
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 C2 2007 January Q7
8 marks Moderate -0.8
    1. Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0° \leq x \leq 360°\), indicating the coordinates of any points where the curve meets the axes. [2]
    2. Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0°\) and \(360°\). [3]
  2. Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(-180°\) and \(180°\). [3]
OCR MEI C2 2006 June Q7
5 marks Easy -1.3
  1. Sketch the graph of \(y = \cos x\) for \(0° \leq x \leq 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leq x \leq 360°\). Label each graph clearly. [3]
  2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leq x \leq 360°\). [2]
OCR MEI C2 2016 June Q5
4 marks Easy -1.3
  1. Fig. 5 shows the graph of a sine function. \includegraphics{figure_5} State the equation of this curve. [2]
  2. Sketch the graph of \(y = \sin x - 3\) for \(0° \leq x \leq 450°\). [2]
Edexcel C2 Q4
8 marks Moderate -0.8
    1. Sketch the curve \(y = \sin (x - 30)°\) for \(x\) in the interval \(-180 \leq x \leq 180\).
    2. Write down the coordinates of the turning points of the curve in this interval. [4]
  2. Find all values of \(x\) in the interval \(-180 \leq x \leq 180\) for which $$\sin (x - 30)° = 0.35,$$ giving your answers to 1 decimal place. [4]
Edexcel C2 Q4
6 marks Moderate -0.3
  1. Sketch on the same diagram the graphs of \(y = \sin 2x\) and \(y = \tan \frac{x}{2}\) for \(x\) in the interval \(0 \leq x \leq 360°\). [4]
  2. Hence state how many solutions exist to the equation $$\sin 2x = \tan \frac{x}{2},$$ for \(x\) in the interval \(0 \leq x \leq 360°\) and give a reason for your answer. [2]
OCR C2 Q6
8 marks Moderate -0.8
$$f(x) = \cos 2x, \quad 0 \leq x \leq \pi.$$
  1. Sketch the curve \(y = f(x)\). [2]
  2. Write down the coordinates of any points where the curve \(y = f(x)\) meets the coordinate axes. [3]
  3. Solve the equation \(f(x) = 0.5\), giving your answers in terms of \(\pi\). [3]
OCR MEI C2 Q3
5 marks Easy -1.3
  1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2x\) for values of \(x\) from \(0\) to \(2\pi\). [3]
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\). [2]
OCR MEI C2 Q5
5 marks Easy -1.2
  1. Sketch the graph of \(y = \cos x\) for \(0° \leqslant x \leqslant 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leqslant x \leqslant 360°\). Label each graph clearly. [3]
  2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leqslant x \leqslant 360°\). [2]
OCR MEI C2 Q6
5 marks Moderate -0.8
  1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2\pi\). [2]
  2. Solve the equation \(2 \sin \theta = -1\) for \(0 \leqslant \theta \leqslant 2\pi\). Give your answers in the form \(k\pi\). [3]
OCR MEI C2 Q7
4 marks Moderate -0.8
Sketch the curve \(y = \sin x\) for \(0° \leqslant x \leqslant 360°\). Solve the equation \(\sin x = -0.68\) for \(0° \leqslant x \leqslant 360°\). [4]
OCR MEI C2 Q8
5 marks Moderate -0.8
  1. Sketch the graph of \(y = \tan x\) for \(0° \leqslant x \leqslant 360°\). [2]
  2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0° \leqslant x \leqslant 360°\). [3]
OCR MEI C2 Q9
4 marks Moderate -0.8
Sketch the graph of \(y = \sin x\) for \(0° \leqslant x \leqslant 360°\). Solve the equation \(\sin x = -0.2\) for \(0° \leqslant x \leqslant 360°\). [4]
OCR MEI C3 2011 June Q9
18 marks Standard +0.3
Fig. 9 shows the curve \(y = f(x)\). The endpoints of the curve are P \((-\pi, 1)\) and Q \((\pi, 3)\), and \(f(x) = a + \sin bx\), where \(a\) and \(b\) are constants. \includegraphics{figure_9}
  1. Using Fig. 9, show that \(a = 2\) and \(b = \frac{1}{2}\). [3]
  2. Find the gradient of the curve \(y = f(x)\) at the point \((0, 2)\). Show that there is no point on the curve at which the gradient is greater than this. [5]
  3. Find \(f^{-1}(x)\), and state its domain and range. Write down the gradient of \(y = f^{-1}(x)\) at the point \((2, 0)\). [6]
  4. Find the area enclosed by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \pi\). [4]
OCR MEI C3 2014 June Q4
7 marks Standard +0.3
Fig. 4 shows the curve \(y = f(x)\), where $$f(x) = a + \cos bx, \quad 0 \leq x \leq 2\pi,$$ and \(a\) and \(b\) are positive constants. The curve has stationary points at \((0, 3)\) and \((2\pi, 1)\). \includegraphics{figure_4}
  1. Find \(a\) and \(b\). [2]
  2. Find \(f^{-1}(x)\), and state its domain and range. [5]
OCR FP3 2008 January Q7
11 marks Challenging +1.3
    1. Verify, without using a calculator, that \(\theta = \frac{1}{8}\pi\) is a solution of the equation \(\sin 6\theta = \sin 2\theta\). [1]
    2. By sketching the graphs of \(y = \sin 6\theta\) and \(y = \sin 2\theta\) for \(0 < \theta < \frac{1}{2}\pi\), or otherwise, find the other solution of the equation \(\sin 6\theta = \sin 2\theta\) in the interval \(0 < \theta < \frac{1}{2}\pi\). [2]
  2. Use de Moivre's theorem to prove that $$\sin 6\theta = \sin 2\theta (16 \cos^4 \theta - 16 \cos^2 \theta + 3).$$ [5]
  3. Hence show that one of the solutions obtained in part (i) satisfies \(\cos^2 \theta = \frac{1}{4}(2 - \sqrt{2})\), and justify which solution it is. [3]