sin goes up, cos goes down

The idea

As angle A increases from 0° to 90°, sin A increases from 0 to 1 and cos A decreases from 1 to 0.

Two ratios, opposite trends

You have seen sin 0° = 0 and sin 90° = 1. You have seen cos 0° = 1 and cos 90° = 0.

Between 0° and 90° there are the special angles 30°, 45° and 60°. As A grows, the opposite side BC grows and the adjacent side AB shrinks.

As A grows from 0° to 90°, what happens to sin A?

• It increases from 0 to 1.
• It decreases from 1 to 0.
• It stays the same.

Sweep the angle

Move the dial to each special angle. The bars show sin A and cos A. The table keeps the values you visit.

Up and down

As A increases from 0° to 90°, sin A increases from 0 to 1 and cos A decreases from 1 to 0.

This is Table 8.1 of your book for sin and cos. Going right, sin A rises and cos A falls. They are equal at 45°.

• A | 0° | 30° | 45° | 60° | 90°
• sin A | 0 | 1/2 | 1/√2 | √3 / 2 | 1
• cos A | 1 | √3 / 2 | 1/√2 | 1/2 | 0

Notes

From 0° to 90°, sin A increases from 0 to 1 and cos A decreases from 1 to 0.

Check yourself

Which is larger, sin 60° or sin 30°?

Which is larger, cos 60° or cos 30°?

Find the value of sin 90° − sin 0°.

Answer: 1

sin 90° − sin 0° = 1 − 0 = 1.

At which of these angles is sin A equal to cos A?

• sin 60° — correct. Yes! sin A increases with A. sin 60° = √3 / 2 is more than sin 30° = 1/2.
• sin 30°. sin A grows as A grows. The larger angle has the larger sin.
• They are equal. They are not equal. sin 30° = 1/2 and sin 60° = √3 / 2.
• cos 60°. cos A falls as A grows. The smaller angle has the larger cos.
• cos 30° — correct. Yes! cos A decreases with A. cos 30° = √3 / 2 is more than cos 60° = 1/2.
• They are equal. They are not equal. cos 30° = √3 / 2 and cos 60° = 1/2.
• 30°. sin 30° = 1/2 but cos 30° = √3 / 2.
• 45° — correct. Yes! sin 45° = cos 45° = 1/√2. This is where the rising sin meets the falling cos.
• 60°. sin 60° = √3 / 2 but cos 60° = 1/2.