Graphs of SINE & COSINE of Angles – SS1 Mathematics Lesson Note (Second Term)

The graphs of sine and cosine show how sin θ and cos θ change as the angle θ increases from 0° to 360°, and both are drawn from the unit circle.

Behavioural objectives

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

1. describe a unit circle;
2. state that sin θ = y and cos θ = x for a point P(x, y) on the unit circle;
3. explain how the graphs of sin θ and cos θ are drawn from the unit circle;
4. state the range of values of sin θ and cos θ;
5. give the values of sin θ and cos θ at 0°, 90°, 180°, 270° and 360°.

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.

• Radius OP: the line from the origin O to a point P on the circle, with length |OP| = 1 unit.
• Point P(x, y): a point that moves round the circumference as the angle changes.
• Angle θ: the angle OP makes with the positive x-axis (Ox), measured anticlockwise.

How are sin θ and cos θ read from the unit circle?

For a point P(x, y) on the unit circle, sin θ is the y-value and cos θ is the x-value.

• Sine: sin θ = y/1 = y, because |OP| = 1.
• Cosine: cos θ = x/1 = x, because |OP| = 1.
• Meaning: the height of P above the x-axis measures sin θ, and its distance from the y-axis measures cos θ.

Signs of sin θ and cos θ in each quadrant

How do you draw the graph of sin θ?

The graph of sin θ is a wave-shaped curve got by plotting the y-value of P against the angle θ.

1. Draw a unit circle and mark P at angles 0°, 30°, 60° and so on up to 360°.
2. Read the y-value of P at each angle, or use a calculator or tables.
3. Draw a horizontal axis for θ (0° to 360°) and a vertical axis for sin θ (-1 to 1).
4. Plot each point (θ, sin θ).
5. Join the points with a smooth curve, not with straight rulers.

How do you draw the graph of cos θ?

The graph of cos θ is a wave-shaped curve got by plotting the x-value of P against the angle θ.

1. Use the same unit circle and the same angles.
2. Read the x-value of P at each angle.
3. Use the same axes: θ across, cos θ up and down.
4. Plot each point (θ, cos θ).
5. Join the points with a smooth curve.

Table of values at 30° intervals (to 2 decimal places)

What should you note about the sine and cosine curves?

The two curves are smooth waves that stay between -1 and 1.

• Range: all values of sin θ and cos θ lie between -1 and +1.
• Shape: the sine and cosine curves have the same shape but different starting points.
• Sine start: the sine curve starts at 0, rises to 1 at 90°, falls to -1 at 270° and returns to 0 at 360°.
• Cosine start: the cosine curve starts at 1, falls to -1 at 180° and returns to 1 at 360°.
• Symmetry: each curve is symmetrical about its peak (high point) and its trough (low point).
• Two angles: for 0° ≤ θ < 360°, each value between -1 and 1 is given by two angles, for example sin 30° = sin 150° = 0.5.
• Single angles: the highest and lowest values occur at one angle each in this range, for example sin θ = 1 only at 90°.

Values of sin θ and cos θ at the quarter turns

What is the difference between the sine curve and the cosine curve?

The sine curve plots the y-value of P against θ, and the cosine curve plots the x-value of P against θ.

Evaluation

1. What is a unit circle?
2. For a point P(x, y) on the unit circle, which coordinate gives sin θ and which gives cos θ?
3. Between which two values do all values of sin θ and cos θ lie?
4. Without looking at your notes, write the values of sin θ and cos θ at 0°, 90°, 180°, 270° and 360°.

Assignment

1. Using the table of values at 30° intervals, draw the graph of sin θ for 0° ≤ θ ≤ 360° on graph paper. On the same axes, draw the graph of cos θ. Label the highest and lowest points of each curve.