The diagram shows the graph of \(y=3\cos2x-1\) for \(0^\circ\leqslant x\leqslant360^\circ\).

(a) Write down the amplitude and period of the graph.

(b) Sketch the graph, showing its key points.

In this question, all angles are in radians. (a) Write down the period of \(5 \tan \left(\frac{x}{4}\right)+1\).

(b) On the axes, sketch the graph of \(y=5 \tan \left(\frac{x}{4}\right)+1\) for \(-2 \pi \leqslant x \leqslant 4 \pi\).

State the intercept with the \(y\)-axis. Show clearly the positions of any asymptotes.

The diagram shows the curve \(y=a\cos bx+c\) for \(-180^\circ\leqslant x\leqslant180^\circ\).

It is given that \(a\), \(b\) and \(c\) are integers.

Find the values of \(a\), \(b\) and \(c\).

The diagram shows part of the graph of \(\mathrm{f}(x)=a \cos b x+c\), where \(a, b\) and \(c\) are constants. Given that \(\mathrm{f}(x)\) has a period of \(960^{\circ}\), find the values of \(a, b\) and \(c\).

The curve \(y=a \cos b x+c\), where \(a, b\) and \(c\) are integers, passes through the points \(\left(-\frac{\pi}{6},-2\right)\) and \(\left(\frac{\pi}{9}, \frac{1}{2}\right)\). The curve has a period of \(\frac{2 \pi}{3}\). (a) Find the values of \(a, b\) and \(c\). (b) Find the least value of \(y\) on the curve for \(0 \leqslant x \leqslant \frac{\pi}{2}\), and state the value of \(x\) at which this occurs.

On the axes, sketch the graph of \(y=4+5 \sin \frac{\theta}{2}\), for \(-360^{\circ} \leqslant \theta \leqslant 360^{\circ}\). State the intercept with the \(y\)-axis.

Given that \(y=2+4 \cos 3 \theta\), for \(-120^{\circ} \leqslant \theta \leqslant 120^{\circ}\), (a) write down the amplitude of \(y\)

(b) write down the period of \(y\).

(c) On the axes, sketch the graph of \(y\).

(a) The diagram shows the graph of \(y=a \cos b x+c\), for \(-360^{\circ} \leqslant x \leqslant 360^{\circ}\), where \(a, b\) and \(c\) are constants. Find the values of \(a, b\) and \(c\).

(b) The line \(y=p\) is a tangent to the curve \(y=3-2 \sin 6 \theta\). Write down the possible values of \(p\).

The diagram shows the graph of \(\quad y=a \sin b x+c\) for \(-360^{\circ} \leqslant x \leqslant 360^{\circ}\), where \(a, b\) and \(c\) are constants. Find the values of \(a, b\) and \(c\).

The diagram shows the graph of \(y=a \sin b x+c\), for \(-320^{\circ} \leqslant x \leqslant 320^{\circ}\), where \(a, b\) and \(c\) are constants. Find the values of \(a, b\) and \(c\).

The diagram shows the graph of \(y=a\cos bx+c\). Find the values of the constants \(a\), \(b\) and \(c\).

(a) Write down the period, in radians, of \(3\tan\frac{\theta}{2}-3\).

(b) On the axes, sketch the graph of \(y=3\tan\frac{\theta}{2}-3\) for \(-\pi\leq\theta\leq\pi\), stating the coordinates of the points where the graph meets the axes.

The function \(g\) is defined for \(0^\circ\leq x\leq120^\circ\) by

\(g(x)=2+4\cos6x.\)

(a) Sketch the graph of \(y=g(x)\).

(b) State the amplitude of \(g\).

(c) State the period of \(g\).

The diagram shows part of the graph of

\(y=a\cos\left(\frac{x}{b}\right)+c,\)

where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).

The function \(g\) is defined by

\(g(x)=5\sin\left(\frac{3x}{4}\right)-2\)

for all values of \(x\).

(a) Write down the amplitude of \(g\).

(b) Write down the period of \(g\) in degrees.

(c) Sketch the graph of \(y=g(x)\), for \(-180^\circ\leq x\leq180^\circ\).

On the axes, draw the graph of

\(y=2\sin\frac{x}{3}-1\)

for \(-360^\circ \leq x \leq 360^\circ\).

The curve with equation \(y=a\sin bx+c\), where \(a\), \(b\) and \(c\) are constants, passes through the points \((4\pi,11)\) and \(\left(-\frac{4\pi}{3},5\right)\). It is given that \(a\sin bx+c\) has period \(16\pi\).

(a) Find the exact values of \(a\), \(b\) and \(c\).

(b) Using your answer to part (a), find the coordinates of the minimum point on the curve for \(0\le x\le16\pi\).

Sketch the graph of

\(y=\left|4\cos 2x\right|\)

for \(0\leq x\leq \pi\), giving the coordinates of the points where the graph meets the axes.

The diagram shows the graph of \(y=a\sin bx+c\), where \(a\), \(b\) and \(c\) are integers, for \(-180^{\circ}\le x\le 180^{\circ}\). Find the values of \(a\), \(b\) and \(c\).

The graph of \(y=a\tan bx+c\) has asymptotes at \(x=-4\pi\) and \(x=4\pi\), and passes through the points \(P(0,3)\) and \(Q(2\pi,7)\).

(a) Find the period of the graph.

(b) Find the values of \(a\), \(b\) and \(c\).

(a) On the axes, sketch the graph of

\(y=5\sin\frac{x}{2}+1\)

for \(-2\pi\leq x\leq 2\pi\).

(b) Write down the amplitude of \(5\sin\frac{x}{2}+1\).

(c) Write down the period of \(5\sin\frac{x}{2}+1\).

The diagram shows the graph of \(y=a\sin bx+c\), where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).

On the axes, sketch the graph of

\(y=4\sin3x-2\)

for \(-\frac{\pi}{3}\leq x\leq\frac{\pi}{3}\).

The diagram shows the graph of \(y=a\sin bx+c\), where \(x\) is in radians and \(-2\pi\leqslant x\leqslant2\pi\), and where \(a\), \(b\) and \(c\) are positive constants.

Find the value of each of \(a\), \(b\) and \(c\).

The graph shows the curve

\(y=a\cos bx+c,\)

for \(0\leq x\leq2.8\), where \(a\), \(b\) and \(c\) are constants and \(x\) is in radians. The curve meets the \(y\)-axis at \((0,3)\) and the \(x\)-axis at the point \(P\) and point \(R\left(\frac{5\pi}{6},0\right)\).

The curve has a minimum at point \(Q\). The period of \(a\cos bx+c\) is \(\pi\) radians.

(a) Find the value of each of \(a\), \(b\) and \(c\).

(b) Find the coordinates of \(P\).

(c) Find the coordinates of \(Q\).

The graph of

\(y=a+2\tan bx,\)

where \(a\) and \(b\) are constants, passes through the point \((0,-4)\) and has period \(480^\circ\).

(a) Find the value of \(a\) and of \(b\).

(b) On the axes, sketch the graph of \(y\) for values of \(x\) between \(0^\circ\) and \(480^\circ\).

The function \(\mathrm f\) is defined, for \(0^\circ\leq x\leq810^\circ\), by

\(\mathrm f(x)=-2+\cos\frac{2x}{3}.\)

(a) Write down the amplitude of \(\mathrm f\).

(b) Find the period of \(\mathrm f\).

(c) On the axes, sketch the graph of \(y=\mathrm f(x)\).

The diagram shows the graph of \(y=a\sin\frac{x}{b}+c\) for \(-360^\circ\leq x\leq 360^\circ\), where \(a\), \(b\) and \(c\) are integers.

(a) Write down the period of \(a\sin\frac{x}{b}+c\).

(b) Find the value of \(a\), of \(b\) and of \(c\).

Find the possible values of the constant \(c\) for which the line \(y=c\) is a tangent to the curve

\(y=5\sin\frac{x}{3}+4.\)

(a) The curve has equation

\(y=a\cos bx+c,\)

where \(a\), \(b\) and \(c\) are integers. Find the values of \(a\), \(b\) and \(c\).

(b) Another curve has equation

\(y=2\sin3x+4.\)

Write down:

(i) the amplitude,

(ii) the period in radians.

The diagram shows the graph of \(y=f(x)\), where

\(f(x)=a\cos bx+c\)

for \(0\leq x\leq \frac{8\pi}{3}\).

Explain why \(f\) is a function, state the range of \(f\), and find the values of \(a\), \(b\) and \(c\).

(a) Write down the period of

\(2\cos\frac{x}{3}-1.\)

(b) Sketch the graph of

\(y=2\cos\frac{x}{3}-1\)

for \(-360^\circ\leq x\leq360^\circ\).

(a) Solve

\(\tan3x=-1\)

for \(-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}\) radians, giving your answers in terms of \(\pi\).

(b) Use your answers to part (a) to sketch the graph of

\(y=4\tan3x+4\)

for \(-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}\) radians. Show the coordinates of the points where the curve meets the axes.

(a) The curve \(y=a\sin bx+c\) has a period of \(180^\circ\), an amplitude of \(20\) and passes through the point \((90^\circ,-3)\). Find the value of each of the constants \(a\), \(b\) and \(c\).

(b) The function \(g\) is defined, for \(-135^\circ\leq x\leq135^\circ\), by

\(g(x)=3\tan\frac{x}{2}-4.\)

Sketch the graph of \(y=g(x)\), stating the coordinates of the point where the graph crosses the y-axis.

(a) Write down the amplitude of \(1+4\cos\frac{x}{3}\).

(b) Write down the period of \(1+4\cos\frac{x}{3}\).

(c) Sketch the graph of \(y=1+4\cos\frac{x}{3}\) for \(-180^\circ\leq x\leq180^\circ\).

The function \(\mathrm{f}\) is given by

\(\mathrm{f}(x)=2\cos\frac{x}{3}-1.\)

(a) Write down the amplitude of \(\mathrm{f}\).

(b) Write down the period of \(\mathrm{f}\).

(c) Sketch the graph of \(y=\mathrm{f}(x)\) for \(-\pi\leq x\leq3\pi\).

The function \(f\) is defined, for \(0^\circ\leq x\leq360^\circ\), by \(f(x)=a+b\sin cx\), where \(a\), \(b\) and \(c\) are constants with \(b\gt 0\) and \(c\gt 0\). The graph of \(y=f(x)\) meets the \(y\)-axis at the point \((0,-1)\), has a period of \(120^\circ\) and an amplitude of \(5\).

(i) Sketch the graph of \(y=f(x)\).

(ii) Write down the value of each of the constants \(a\), \(b\) and \(c\).

(i) Write down the amplitude of \(4\sin 3x-1\).

(ii) Write down the period of \(4\sin 3x-1\).

(iii) Sketch the graph of \(y=4\sin 3x-1\) for \(-90^\circ\leq x\leq90^\circ\).

The function \(f\) is defined, for \(0^\circ\leq x\leq360^\circ\), by \(f(x)=4+3\sin2x\).

(i) Sketch the graph of \(y=f(x)\).

(ii) State the period of \(f\).

(iii) State the amplitude of \(f\).

(i) Sketch the graph of

\(y=2\cos3x-1\quad\text{for }-90^\circ\leq x\leq90^\circ.\)

(ii) Write down the amplitude of \(2\cos3x-1\).

(iii) Write down the period of \(2\cos3x-1\).

(i) Sketch the graph of

\(y=5\cos4x-3\quad\text{for }-90^\circ\leq x\leq90^\circ.\)

(ii) Write down the amplitude of \(y\).

(iii) Write down the period of \(y\).

The figure shows part of the graph of \(y=p+q\cos rx\). Find the value of each of the integers \(p\), \(q\) and \(r\).

(a) (i) State the amplitude of \(15\sin2x-5\).

(ii) State the period of \(15\sin2x-5\).

(b) The diagram shows the graph of \(y=f(x)\), where \(f(x)\) is a trigonometric function.

(i) Write down two possible expressions for the trigonometric function \(f(x)\).

(ii) State the number of solutions of the equation \(f(x)=1\) for \(-180^\circ\leq x\leq180^\circ\).

(i) The curve

\(y=a+b\sin cx\)

has an amplitude of \(4\) and a period of \(\dfrac{\pi}{3}\). Given that the curve passes through the point \(\left(\dfrac{\pi}{12},2\right)\), find the value of each of the constants \(a\), \(b\) and \(c\).

(ii) Using your values of \(a\), \(b\) and \(c\), sketch the graph of \(y=a+b\sin cx\) for \(0\leq x\leq\pi\) radians.

It is given that \(y=1+\tan3x\).

(i) State the period of \(y\).

(ii) Sketch the graph of \(y=1+\tan3x\) for \(0^\circ\leqslant x\leqslant180^\circ\).

(i) The curve

\(y=a+b\sin cx\)

has an amplitude of \(4\) and a period of \(\dfrac{\pi}{3}\). Given that the curve passes through the point \(\left(\dfrac{\pi}{12},2\right)\), find the value of each of the constants \(a\), \(b\) and \(c\).

(ii) Using your values of \(a\), \(b\) and \(c\), sketch the graph of \(y=a+b\sin cx\) for \(0\leq x\leq\pi\) radians.

(a) Sketch the graph of \(y=3\cos2x-1\), for \(0^\circ\leq x\leq360^\circ\).

(b) Given that \(y=4\sin6x\), write down

(i) the amplitude of \(y\),

(ii) the period of \(y\).

The diagram shows part of the curve \(y=a+4\cos bx\), where \(a\) and \(b\) are positive constants. The curve meets the \(y\)-axis at the point \((0,6)\) and the \(x\)-axis at the point \(\left(\dfrac{\pi}{6},0\right)\). The curve meets the \(x\)-axis again at the point \(P\) and has a minimum at the point \(M\).

(i) Find the value of \(a\) and of \(b\).

Using your values of \(a\) and \(b\), find

(ii) the exact coordinates of \(P\),

(iii) the exact coordinates of \(M\).

Given that \(y=3+4\cos 9x\), write down

(i) the amplitude of \(y\),

(ii) the period of \(y\).

(a) Given that \(y=7\cos10x-3\), where the angle \(x\) is measured in degrees, state

(i) the period of \(y\),

(ii) the amplitude of \(y\).

(b) Find the equation of the curve shown, in the form \(y=ag(bx)+c\), where \(g(x)\) is a trigonometric function and \(a\), \(b\) and \(c\) are integers to be found.

The graph of \(y=a\sin(bx)+c\) has an amplitude of \(4\), a period of \(\dfrac{\pi}{3}\), and passes through the point \(\left(\dfrac{\pi}{12},2\right)\). Find the value of each of the constants \(a\), \(b\), and \(c\).

Given that \(y=2\operatorname{sec}^2\theta\) and \(x=\tan\theta-5\), express \(y\) in terms of \(x\).

The graph of \(y=a\cos(bx)+c\) has an amplitude of \(3\), a period of \(\dfrac{\pi}{4}\), and passes through the point \(\left(\dfrac{\pi}{12},\dfrac52\right)\). Find the value of each of the constants \(a\), \(b\), and \(c\).