Find stationary points - trigonometric functions

Find stationary points of curves involving trigonometric functions (e.g., y = sin²(2x)cos(x), y = sin(x)cos(2x), y = x + cos(2x)). Often requires trigonometric identities and solving transcendental equations.

10 questions · Standard +0.4

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CAIE P2 2024 June Q1
3 marks Standard +0.3
1 A curve has equation \(\mathrm { y } = 2 \tan \mathrm { x } - 5 \sin \mathrm { x }\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinate of the stationary point of the curve. Give your answer correct to 3 significant figures.
CAIE P2 2006 June Q3
7 marks Moderate -0.3
3 The equation of a curve is \(y = x + 2 \cos x\). Find the \(x\)-coordinates of the stationary points of the curve for \(0 \leqslant x \leqslant 2 \pi\), and determine the nature of each of these stationary points.
CAIE P3 2011 June Q8
10 marks Standard +0.8
8
\includegraphics[max width=\textwidth, alt={}, center]{5b219e1c-e5a0-4f75-910d-fca9761e5088-3_435_895_799_625} The diagram shows the curve \(y = 5 \sin ^ { 3 } x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \cos x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2013 June Q9
10 marks Challenging +1.2
9
\includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-3_307_601_1553_772} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \sin x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2015 June Q4
7 marks Standard +0.3
4 The equation of a curve is $$y = 3 \cos 2 x + 7 \sin x + 2$$ Find the \(x\)-coordinates of the stationary points in the interval \(0 \leqslant x \leqslant \pi\). Give each answer correct to 3 significant figures.
CAIE P3 2015 June Q3
6 marks Standard +0.3
3 A curve has equation \(y = \cos x \cos 2 x\). Find the \(x\)-coordinate of the stationary point on the curve in the interval \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
CAIE P3 2016 June Q5
6 marks Standard +0.3
5 The curve with equation \(y = \sin x \cos 2 x\) has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.
CAIE P3 2005 November Q3
7 marks Standard +0.3
3 The equation of a curve is \(y = x + \cos 2 x\). Find the \(x\)-coordinates of the stationary points of the curve for which \(0 \leqslant x \leqslant \pi\), and determine the nature of each of these stationary points.
CAIE P3 2020 June Q4
6 marks Standard +0.3
4 A curve has equation \(y = \cos x \sin 2 x\).
Find the \(x\)-coordinate of the stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
OCR C4 2013 June Q4
6 marks Standard +0.3
4 The equation of a curve is \(y = \cos 2 x + 2 \sin x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the stationary points on the curve for \(0 < x < \pi\).