The unit circle and trigonometric equations Problem set A One question at a time

A point P lies on a circle with centre O at the origin and radius 8 cm. The line OP makes an angle of 205° with the positive x-axis, measured anticlockwise. Find the coordinates of P.

• \(x=r\cos\theta=8\cos205^\circ=-7.250\)
• \(y=r\sin\theta=8\sin205^\circ=-3.381\)
• \(P=(-7.25,\;-3.38)\)

The angle θ is obtuse and \(\sin\theta=\frac{8}{17}\). Find the exact value of \(\cos\theta\).

• \(\cos^2\theta=1-\left(\frac{8}{17}\right)^2=\frac{225}{289}\)
• \(\cos\theta=\pm\frac{15}{17}\)
• θ is obtuse, so it is in the second quadrant and cosine is negative, so \(\cos\theta=-\frac{15}{17}\)

The angle θ is between 180° and 270° and \(\cos\theta=-0.6\).

1. (a) Find \(\sin\theta\).

   • \(\sin^2\theta=1-(-0.6)^2=0.64\)
   • \(\sin\theta=\pm0.8\); θ is in the third quadrant, so \(\sin\theta=-0.8\)

2. (b) Find \(\tan\theta\).

   • \(\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-0.8}{-0.6}=\frac43\)
   • Tangent is positive in the third quadrant

Given that sin 24° = 0.407 and cos 24° = 0.914 (3 s.f.), use the unit circle to answer the following.

1. (a) Write down another angle between 0° and 360° with the same sine as 24°.

   • Same sine: \(180^\circ-24^\circ=156^\circ\)

2. (b) Write down another angle between 0° and 360° with the same cosine as 24°.

   • Same cosine: \(360^\circ-24^\circ=336^\circ\)

3. (c) cos 0.8 = 0.697 (radians). Write down another angle between 0 and 2π with the same cosine.

   • Same cosine in radians: \(2\pi-0.8=5.483\)

Solve each equation for \(0^\circ\le x\le360^\circ\).

1. (a) \(\sin x=0.72\)

   • \(x=\sin^{-1}0.72=46.1^\circ\)
   • \(180^\circ-46.1^\circ=133.9^\circ\)

2. (b) \(\cos x=-0.25\)

   • \(x=\cos^{-1}(-0.25)=104.5^\circ\)
   • \(360^\circ-104.5^\circ=255.5^\circ\)

Solve \(4\cos 2x=1\) for \(0\le x\le\pi\). Give the solutions in radians to 3 decimal places.

• Radian mode. Graph \(y=4\cos\,2x\) and \(y=1\) for \(0\le x\le\pi\)
• Intersections: \(x=0.659\), \(x=2.483\)

A Ferris wheel has radius 15 m and its lowest point is 2 m above the ground. A seat is at the lowest point when t = 0, where t is the time in minutes. One turn takes 12 minutes. The height of the seat above the ground is \(h=17-15\cos\left(\frac{\pi t}{6}\right)\) metres, for \(0\le t\le12\).

1. (a) Find the height of the seat when t = 2.

   • \(h=17-15\cos\frac{2\pi}{6}=\) \(9.500\)
   • 9.50 m

2. (b) Find the times when the seat is 20 m above the ground.

   • Graph the model and the line in radian mode
   • \(t=3.385\) and \(t=8.615\)

3. (c) Find for how long in one turn the seat is more than 20 m above the ground.

   • \(8.615-3.385=5.231\)
   • 5.23

A point P moves round a circle with centre the origin and radius 12 cm. At t = 0 it is at (12, 0). It moves anticlockwise through 0.4 radians every second. Use radian mode.

1. (a) Find the coordinates of P after 5 seconds.

   • Angle after 5 s: \(\theta=0.4\times5=2\) radians
   • \(x=12\cos2=-4.994,\;y=12\sin2=10.912\)

2. (b) Find the first time at which the y-coordinate of P is 9 cm.

   • \(12\sin(0.4t)=9\), so \(\sin(0.4t)=0.75\)
   • \(0.4t=\sin^{-1}0.75=0.8481\), \(t=2.120\) s

The point P(−0.6, y) lies on the unit circle, in the second quadrant. The angle θ is measured anticlockwise from the positive x-axis, with 0 ≤ θ ≤ 2π.

1. (a) Find y.

   • \(x^2+y^2=1\), so \(y^2=1-0.36=0.64\)
   • Second quadrant: y is positive, \(y=0.8\)

2. (b) Find θ in radians.

   • \(\cos\theta=-0.6\)
   • \(\theta=\cos^{-1}(-0.6)=2.2143\) (radian mode); this is between 0 and π, as it should be

3. (c) Find tan θ.

   • \(\tan\theta=\frac{0.8}{-0.6}=-\frac43\)

Consider the equation \(3\sin x=2\cos x\), for \(0^\circ\le x\le360^\circ\).

1. (a) Show that the equation can be written as \(\tan x=\frac{2}{3}\).

   • Divide both sides by \(3\cos x\): \(\frac{\sin x}{\cos x}=\frac{2}{3}\)
   • \(\tan x=\frac{2}{3}\)

2. (b) Hence solve the equation.

   • \(x=\tan^{-1}\frac{2}{3}=33.7^\circ\)
   • \(x=33.7^\circ+180^\circ=213.7^\circ\)