Trigonometric Functions II: Graph of SINE & COSINE of Angles – SS2 Further Mathematics Lesson Note (First Term)

The graphs of sine and cosine are wave-shaped curves drawn by plotting the y- and x-coordinates of a point P on a unit circle against the angle θ, and both lie between +1 and -1 and repeat every 360°.

Behavioural objectives

By the end of the lesson, students should be able to:

1. describe a unit circle and state why the coordinates of P give cos θ and sin θ;
2. draw and describe the graphs of sin θ and cos θ;
3. read values of sin θ and angles from the graph;
4. state the values of sin θ and cos θ at 0°, 90°, 180°, 270° and 360°;
5. write the sine, cosine and tangent of 30°, 45° and 60° in surd form and use them to find related values such as sin 135°.

What is a unit circle?

A unit circle is a circle of radius 1 unit drawn on a Cartesian plane with its centre at the origin O.

• Point P: a point (x, y) on the circumference, so |OP| = 1 unit.
• Angle θ: the angle that OP makes with the x-axis (Ox). It changes as P moves round the circle.
• Sine: sin θ = y/1 = y, so the y-coordinate of P measures sin θ.
• Cosine: cos θ = x/1 = x, so the x-coordinate of P measures cos θ.

How are the graphs of sine and cosine drawn?

The graph of sin θ is drawn by plotting the y-coordinate of P against the matching angle θ.

Steps for drawing the sine curve

1. Draw a unit circle and mark P at θ = 30°, 60°, 90°, …, 360°.
2. Read the y-coordinate of P for each angle. This is sin θ.
3. Draw a horizontal θ axis and a vertical axis for sin θ.
4. Plot each pair (θ, sin θ), using vertical dotted lines to match each angle to its value.
5. Join the points with a smooth curve.

Steps for drawing the cosine curve

1. Use the x-coordinate of P at the same angles. This is cos θ.
2. Plot each pair (θ, cos θ) and join the points with a smooth curve.
3. Continue past 360°. Both curves repeat themselves as θ increases.

What are the properties of the sine and cosine curves?

The sine and cosine curves are waves with the same shape, the same range and the same period.

• Range: all values of sin θ and cos θ lie between +1 and -1.
• Period: each curve repeats after 360°.
• Same shape: the two curves differ only in their starting points.
• Symmetry: each curve is symmetrical about its peak (high point) and trough (low point).
• Two angles per value: for 0° ≤ θ < 360°, most values of sin θ or cos θ occur at two angles.
• Exceptions: the peak and trough values occur at one angle only: sin θ = 1 at 90°, sin θ = -1 at 270°, cos θ = 1 at 0°, cos θ = -1 at 180°.

Sine curve and cosine curve compared

Values at the quarter turns

Example: reading values from the graph

Referring to the graph on page 211 of New General Mathematics Book 1:

1. Find the value of sin 252°.
2. Solve the equation 5 sin θ = 4.

Solution

1. On the θ axis, each small square represents 6°. From construction (a) on the graph, sin 252° = -0.95.
2. If 5 sin θ = 4, then sin θ = 4/5 = 0.8. From construction (b) on the graph, when sin θ = 0.8, θ is about 54° or 126°.

What is the graph of tan θ?

The graph of tan θ is drawn by taking values from a unit circle, using the tangent line at X.

1. Draw a tangent to the unit circle at X, where the circle meets the x-axis.
2. Draw a radius from O and extend it to meet the tangent at T.
3. Read the y-coordinate of T. It measures tan θ, where θ is the angle the radius makes with OX.
4. Plot tan θ against θ for many angles and join the points.

• Undefined values: tan θ is not defined at θ = 90° and 270°.
• Link to sine and cosine: tan θ = sin θ / cos θ, and cos θ = 0 at 90° and 270°.
• Period: the tan curve repeats every 180°.

What are the ratios of the special angles 45°, 30° and 60°?

The special angles 30°, 45° and 60° have exact trigonometric ratios, found from two simple triangles and written in surd form.

What are tan, sin and cos of 45°?

1. Let ABC be a triangle with a right angle at B, where |AB| = |BC| = 1 unit.
2. By Pythagoras’ theorem, |AC|² = 1² + 1² = 2, so |AC| = √2.
3. Angle A is 45°, so read off the ratios.

• tan 45°: 1.
• sin 45°: 1/√2 = √2/2.
• cos 45°: 1/√2 = √2/2.

What are tan, sin and cos of 30° and 60°?

1. Let ABC be an equilateral triangle with sides 2 units long.
2. Draw the altitude AD, so |BD| = |DC| = 1 unit.
3. In triangle ABD, |AB|² = |AD|² + |BD|² (Pythagoras’ theorem).
4. 2² = |AD|² + 1², so |AD|² = 3 and |AD| = √3 units.
5. Angle B = 60° and angle BAD = 30°. Read off the ratios.

• tan 60°: √3.
• sin 60°: √3/2.
• cos 60°: ½ = 0.50.
• tan 30°: 1/√3.
• sin 30°: ½ = 0.50.
• cos 30°: √3/2.

Table of special angles

How do we find the ratios of angles above 90°?

We reduce the angle to an acute angle, then attach the correct sign from the quadrant.

Example: surd form of related angles

Write the value of each of the following in surd form:

1. sin 135°
2. tan 330°
3. cos 240°

Solution

1. sin 135° = sin (180° – 135°) = sin 45° = 1/√2 = √2/2
2. tan 330° = -tan (360° – 330°) = -tan 30° = -1/√3
3. cos 240° = -cos (240° – 180°) = -cos 60° = -½

Evaluation

1. What is a unit circle?
2. Between which two values do all values of sin θ and cos θ lie?
3. Write down the values of sin 270° and cos 180°.
4. State the angles between 0° and 360° at which tan θ is not defined.
5. Write sin 60°, cos 45° and tan 30° in surd form.

Assignment

1. Using an equilateral triangle with sides of 2 units, show that sin 30° = ½ and cos 30° = √3/2.
2. Write sin 120° in surd form, using the method shown in the example above.
3. Write tan 315° in surd form, using the method shown in the example above.