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 changes from 0° to 360°. They are drawn from a unit circle, a circle of radius 1 unit on a Cartesian plane, where the y-value of point P gives sin θ and the x-value gives cos θ.

Behavioural objectives

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

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

What is a unit circle?

Draw a circle on a Cartesian plane so that its radius, OP, is of length 1 unit. Such a circle is called a unit circle.

The angle θ that OP makes with Ox changes according to the position of P on the circumference of the unit circle. Since P is the point (x, y) and |OP| = 1 unit:

sin θ = y/1 = y

cos θ = x/1 = x

Hence the values of y and x give a measure of sin θ and cos θ respectively.

How do you draw the graphs of sine and cosine?

If the values of θ are taken from the unit circle, they can be used to draw the graph of sin θ. This is done by plotting values of y against corresponding values of θ.

To draw the graph of cos θ, use corresponding values of x and θ. This gives another wave-shaped curve.

What should you note about the sine and cosine curves?

1. All values of sin θ and cos θ lie between +1 and -1.
2. The sine and cosine curves have the same shape but different starting points.
3. Each curve is symmetrical about its peak (high point) and trough (low point). This means that for any value of sin θ there are usually two angles between 0° and 360°. The same is true for cos θ. The only exceptions are at the quarter turns, where sin θ and cos θ have the values given in the table below.

Values of sin θ and cos θ at the quarter turns

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. If sin A = 4/5, what is tan A? A. 2/5 B. 3/5 C. 3/4 D. 1 E. 4/3
2. Use tables to find the value of 8 cos 77°. A. 5.44 B. 6.48 C. 9.12 D. 7.57 E. 1.80
3. If cos θ = sin 33°, find tan θ. A. 1.540 B. 2.64 C. 0.64 D. 1.16 E. 1.32
4. If the diagonal of a square is 8 cm, what is the area of the square? A. 16 cm² B. 2 cm² C. 4 cm² D. 20 cm² E. 10 cm²
5. Calculate the angle which the diagonal in question 4 makes with any of the sides of the square. A. 65° B. 45° C. 35° D. 25° E. 75°
6. From a place 400 m north of X, a student walks eastwards to a place Y which is 800 m from X. What is the bearing of X from Y?
7. In the figure below, the right angles and lengths of sides are as shown. Calculate the value of K.