The unit circle and trigonometric equations Problem set B Worked solutions

A point P lies on a circle with centre O at the origin and radius 6.5 cm. The line OP makes an angle of 310° with the positive x-axis, measured anticlockwise. Find the coordinates of P.

• \(x=r\cos\theta=6.5\cos310^\circ=4.178\)
• \(y=r\sin\theta=6.5\sin310^\circ=-4.979\)
• \(P=(4.18,\;-4.98)\)

The angle θ is between 270° and 360° and \(\cos\theta=\frac{12}{37}\). Find the exact value of \(\sin\theta\).

• \(\sin^2\theta=1-\left(\frac{12}{37}\right)^2=\frac{1225}{1369}\)
• \(\sin\theta=\pm\frac{35}{37}\)
• θ is in the fourth quadrant, where sine is negative, so \(\sin\theta=-\frac{35}{37}\)

The angle θ is between 180° and 270° and \(\sin\theta=-0.28\).

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

   • \(\cos^2\theta=1-(-0.28)^2=0.9216\)
   • \(\cos\theta=\pm0.96\); θ is in the third quadrant, so \(\cos\theta=-0.96\)

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

   • \(\tan\theta=\frac{-0.28}{-0.96}=\frac{7}{24}\)
   • Tangent is positive in the third quadrant

Given that sin 37° = 0.602 and cos 37° = 0.799 (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 37°.

   • Same sine: \(180^\circ-37^\circ=143^\circ\)

2. (b) Write down another angle between 0° and 360° with the same cosine as 37°.

   • Same cosine: \(360^\circ-37^\circ=323^\circ\)

3. (c) cos 1.3 = 0.267 (radians). Write down another angle between 0 and 2π with the same cosine.

   • Same cosine in radians: \(2\pi-1.3=4.983\)

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

1. (a) \(\sin x=0.46\)

   • \(x=\sin^{-1}0.46=27.4^\circ\)
   • \(180^\circ-27.4^\circ=152.6^\circ\)

2. (b) \(\cos x=-0.62\)

   • \(x=\cos^{-1}(-0.62)=128.3^\circ\)
   • \(360^\circ-128.3^\circ=231.7^\circ\)

Solve \(6\sin 2x=2.5\) for \(0\le x\le\pi\). Give the solutions in radians to 3 decimal places.

• Radian mode. Graph \(y=6\sin\,2x\) and \(y=2.5\) for \(0\le x\le\pi\)
• Intersections: \(x=0.215\), \(x=1.356\)

The depth of water in a harbour is \(d=6-2.4\cos\left(\frac{\pi t}{6.2}\right)\) metres, where t is the time in hours after low tide, for \(0\le t\le12.4\).

1. (a) Find the depth of water 2 hours after low tide.

   • \(d=6-2.4\cos\frac{2\pi}{6.2}=\) \(4.730\)
   • 4.73 m

2. (b) Find the times when the depth is 7 m.

   • Graph the model and the line in radian mode
   • \(t=3.948\) and \(t=8.452\)

3. (c) A ship can enter the harbour when the depth is more than 7 m. For how long in one tide cycle can it enter?

   • \(8.452-3.948=4.504\)
   • 4.50

The tip P of a fan blade moves round a circle with centre the origin and radius 25 cm. At t = 0 it is at (25, 0). It moves anticlockwise through 0.7 radians every second. Use radian mode.

1. (a) Find the coordinates of P after 3 seconds.

   • Angle after 3 s: \(\theta=0.7\times3=2.1\) radians
   • \(x=25\cos2.1=-12.621,\;y=25\sin2.1=21.580\)

2. (b) Find the first time at which the y-coordinate of P is 15 cm.

   • \(25\sin(0.7t)=15\), so \(\sin(0.7t)=0.6\)
   • \(0.7t=\sin^{-1}0.6=0.6435\), \(t=0.919\) s

The point P(0.8, y) lies on the unit circle, in the fourth 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 \(y^2=1-0.64=0.36\)
   • Fourth quadrant: y is negative, \(y=-0.6\)

2. (b) Find θ in radians.

   • \(\cos^{-1}0.8=0.6435\), the angle in the first quadrant
   • \(\theta=2\pi-0.6435=5.6397\)

3. (c) Find tan θ.

   • \(\tan\theta=\frac{-0.6}{0.8}=-\frac34\)

Consider the equation \(2\sin x=5\cos x\), for \(0^\circ\le x\le360^\circ\).

1. (a) Show that the equation can be written as \(\tan x=\frac{5}{2}\).

   • Divide both sides by \(2\cos x\): \(\frac{\sin x}{\cos x}=\frac{5}{2}\)
   • \(\tan x=\frac{5}{2}\)

2. (b) Hence solve the equation.

   • \(x=\tan^{-1}\frac{5}{2}=68.2^\circ\)
   • \(x=68.2^\circ+180^\circ=248.2^\circ\)