When the angle shrinks to 0°

The idea

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

Squash the triangle

A ladder leans on a wall. Pull its foot away so the ladder lies flatter and flatter on the ground. The angle A at the foot gets smaller and smaller.

In the right triangle ABC the side BC (opposite to A) gets shorter. The point C comes closer to B. When A is very close to 0°, AC is almost the same length as AB.

When A is very close to 0°, what is sin A = BC / AC close to?

• 0
• 1
• It is very large

Flatten the arm

The arm AC has length 10. Drag C along the dashed arc down to the ground, or use the dial. Watch the six ratios.

The ratios of 0°

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

tan 0° = sin 0° / cos 0° = 0

sec 0° = 1 / cos 0° = 1

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

Notes

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

Check yourself

What is the value of tan 0°?

Why is cosec 0° not defined?

What is the value of sec 0°?

Answer: 1

sec 0° = 1 / cos 0° = 1 / 1 = 1.

Find the value of sin 0° + cos 0°.

Answer: 1

sin 0° + cos 0° = 0 + 1 = 1.

• 0 — correct. Yes! tan 0° = sin 0° / cos 0° = 0 / 1 = 0.
• 1. cos 0° is 1, but tan 0° = sin 0° / cos 0° = 0 / 1 = 0.
• Not defined. The bottom of the fraction is cos 0° = 1, not 0. So tan 0° = 0.
• cosec 0° = 1 / sin 0° = 1 / 0, and we cannot divide by 0. — correct. Yes! sin 0° = 0, so 1 / sin 0° has 0 at the bottom.
• Because cos 0° = 0.. cos 0° is 1, not 0. The problem is sin 0° = 0.
• Because the triangle has no hypotenuse.. The hypotenuse AC is still there. The problem is that BC becomes 0.