Teach Me Maths

Trigonometric Graphs and Ratios of Any Angle

Use the unit circle to find trig ratios beyond acute angles, recognise sine, cosine and tangent graphs, and work with exact values and graph features.

Extend the ratios beyond right-angled triangles

On a circle of radius 1 centred at the origin, the point at angle \theta has coordinates (\cos\theta,\sin\theta). Measure positive angles anticlockwise from the positive x-axis.

Where \cos\theta\ne0:

\tan\theta=\frac{\sin\theta}{\cos\theta}

In quadrant I all three ratios are positive. In quadrant II only sine is positive; in III only tangent is positive; in IV only cosine is positive. A reference angle is the acute angle the radius makes with the horizontal axis. Use it for the size of the ratio, then choose the sign from the quadrant.

Exact values and axis angles

The exact values for 30^\circ, 45^\circ and 60^\circ are:

\begin{array}{c|ccc} \theta & \sin\theta & \cos\theta & \tan\theta \\[0.8em] 30^\circ & \frac12 & \frac{\sqrt3}{2} & \frac1{\sqrt3} \\[0.8em] 45^\circ & \frac{\sqrt2}{2} & \frac{\sqrt2}{2} & 1 \\[0.8em] 60^\circ & \frac{\sqrt3}{2} & \frac12 & \sqrt3 \end{array}

At 0^\circ,90^\circ,180^\circ,270^\circ,360^\circ, the unit-circle points are (1,0),(0,1),(-1,0),(0,-1),(1,0). Thus tangent is undefined at 90^\circ and 270^\circ, where cosine is zero.

Worked example 1

Find \sin150^\circ, \cos150^\circ and \tan150^\circ exactly.

  1. 150^\circ is in quadrant II, with reference angle 180^\circ-150^\circ=30^\circ. Sine is positive; cosine and tangent are negative.

  2. Your turn. Give \sin150^\circ exactly.

    \sin150^\circ=\frac12

  3. Answer: \sin150^\circ=\frac12, \cos150^\circ=-\frac{\sqrt3}{2} and \tan150^\circ=-\frac1{\sqrt3}=-\frac{\sqrt3}{3}

Worked example 2

Find \sin240^\circ, \cos240^\circ and \tan240^\circ exactly.

  1. 240^\circ is in quadrant III, with reference angle 240^\circ-180^\circ=60^\circ. Sine and cosine are negative, but their ratio is positive.

  2. Your turn. Give \cos240^\circ exactly.

    \cos240^\circ=-\frac12

  3. Answer: \sin240^\circ=-\frac{\sqrt3}{2}, \cos240^\circ=-\frac12, \tan240^\circ=\sqrt3

Worked example 3

\theta lies in quadrant II and \cos\theta=-5/13. Find \sin\theta and \tan\theta exactly.

  1. The unit-circle identity gives \sin^2\theta=1-\cos^2\theta=1-25/169=144/169. In quadrant II, sine is positive.

  2. Your turn. Find the positive value of \sin\theta

    \sin\theta=12/13

  3. Answer: \sin\theta=12/13 and \tan\theta=(12/13)/(-5/13)=-12/5

Connect the circle to the graphs

Sine and cosine have period 360° and range from −1 to 1. Sine starts at 0; cosine starts at 1. Tangent has period 180°, can take any real value, and has vertical asymptotes at 90° and 270° in this interval. Do not join its branches through the asymptotes.

Worked example 4

Sketch y=3\sin x for 0^\circ\le x\le360^\circ. State its period, range and key points.

  1. Multiply the usual sine heights by 3. The key points are (0,0),(90,3),(180,0),(270,-3),(360,0), with x measured in degrees.

  2. Your turn. What is the minimum value of y?

    The minimum is -3, reached at x=270^\circ

  4. Answer: period 360^\circ, range -3\le y\le3, and zeros at 0^\circ,180^\circ,360^\circ

Worked example 5

For y=2\cos x-1 on 0^\circ\le x\le360^\circ, find the maximum, minimum and the points where the graph crosses the x-axis.

  1. Because -1\le\cos x\le1, multiplying by 2 then subtracting 1 gives -3\le y\le1

  2. Your turn. What is the maximum value of y?

    The maximum is 1, at x=0^\circ and 360^\circ

  3. At an x-axis crossing, 2\cos x-1=0, so \cos x=1/2. Cosine is positive in quadrants I and IV, giving x=60^\circ,300^\circ
  4. Answer: maximum 1, minimum -3 (at 180^\circ), crossings (60,0) and (300,0) with angles in degrees.

Worked example 6

Describe the graph y=\tan x for 0^\circ\le x\le360^\circ, including its zeros, asymptotes and period. Find \tan225^\circ.

  1. Tangent is zero where sine is zero and cosine is non-zero: x=0^\circ,180^\circ,360^\circ. It is undefined at 90^\circ,270^\circ, where cosine is zero. These are vertical asymptotes.

  2. Your turn. What is the period of tangent in degrees?

    The period is 180^\circ

  3. 225^\circ=180^\circ+45^\circ, so \tan225^\circ=\tan45^\circ=1
  4. Answer: zeros 0^\circ,180^\circ,360^\circ; asymptotes 90^\circ,270^\circ; period 180^\circ; \tan225^\circ=1. Each branch increases, with a break at each asymptote.

Common mistakes

  • Keeping the reference angle’s positive sign. Use the reference angle for size and the quadrant for sign.
  • Swapping the unit-circle coordinates. The coordinates are (\cos\theta,\sin\theta): cosine across, sine up.
  • Treating tangent at 90° as a finite value. \tan90^\circ is undefined because \cos90^\circ=0.
  • Joining across a tangent asymptote. Draw separate branches on either side of each asymptote.
  • Changing the period when scaling vertically. A non-zero multiplier outside the trig function changes heights, not the period.

Now try it: Trigonometric Graphs and Ratios of Any Angle practice questions

More on this topic: Trigonometric Graphs and Ratios of Any Angle worksheet with full solutions