When the angle grows to 90°

The idea

sin 90° = 1 and cos 90° = 0, the values sin A and cos A approach as A nears 90°; so cosec 90° = 1 and cot 90° = 0, while tan 90° and sec 90° are not defined.

Stand the ladder up

Now push the foot of the ladder closer to the wall. The ladder stands more and more upright, so A gets larger and larger.

In the right triangle ABC, the angle C gets smaller and smaller. The side AB (adjacent to A) shrinks. The point A comes closer to B, and AC almost lies along BC.

When A is very close to 90°, what is cos A = AB / AC close to?

• 0
• 1
• It is very large

Stand the arm up

Drag C up the dashed arc until the arm stands straight up, or use the dial. Watch the six ratios.

The ratios of 90°

As A nears 90°, AB nears 0 and AC nears BC. So we define sin 90° = 1 and cos 90° = 0.

cosec 90° = 1 / sin 90° = 1

cot 90° = cos 90° / sin 90° = 0

tan 90° = sin 90° / cos 90° and sec 90° = 1 / cos 90° would need a division by 0, so they are not defined.

Notes

sin 90° = 1, cos 90° = 0, cosec 90° = 1, cot 90° = 0. tan 90° and sec 90° are not defined, because they need a division by 0.

Check yourself

Why is tan 90° not defined?

What is the value of cot 90°?

Answer: 0

cot 90° = cos 90° / sin 90° = 0 / 1 = 0.

Which two ratios are not defined at 90°?

Find the value of sin 90° + cos 90°.

Answer: 1

sin 90° + cos 90° = 1 + 0 = 1.

• tan 90° = sin 90° / cos 90° = 1 / 0, and we cannot divide by 0. — correct. Yes! cos 90° = 0, so the bottom of the fraction is 0.
• Because sin 90° = 0.. sin 90° is 1, not 0. The problem is cos 90° = 0.
• Because 90° is not an acute angle.. The problem is the division by 0, not the name of the angle.
• tan 90° and sec 90° — correct. Yes! Both have cos 90° = 0 at the bottom.
• cot 90° and cosec 90°. cot 90° = 0 and cosec 90° = 1. Both are defined.
• sin 90° and cos 90°. sin 90° = 1 and cos 90° = 0. Both are defined.