Turn the ratio upside down

The idea

cosec A, sec A and cot A are the reciprocals of sin A, cos A and tan A: cosec A = hypotenuse/side opposite, sec A = hypotenuse/side adjacent, cot A = side adjacent/side opposite.

Flip a fraction

You know sin A = opposite / hypotenuse. For the 3-4-5 triangle that is 3/5.

What if we divide the other way round, hypotenuse by opposite? That gives a new ratio.

What is hypotenuse / opposite for the 3-4-5 triangle?

• 5/3
• 3/5
• 4/5

Flip each ratio

Drag C to any triangle. Press ⇅ on a row: the fraction turns upside down and gets a new name. Look at the product line that appears.

Three reciprocals

cosec A = 1/sin A = AC/BC sec A = 1/cos A = AC/AB cot A = 1/tan A = AB/BC

These are the cosecant, secant and cotangent of A. Each is the reciprocal of one ratio you know, so each pair multiplies to 1. Note which pairs go together: sin with cosec, cos with sec, tan with cot.

Notes

cosec A = 1/sin A = AC/BC sec A = 1/cos A = AC/AB cot A = 1/tan A = AB/BC

Check yourself

Which ratio is hypotenuse / opposite?

In the 3-4-5 triangle (BC = 3, AB = 4, AC = 5), what is cot A? Give a decimal.

Answer: 1.33

cot A = AB/BC = 4/3, which is about 1.33.

If sin A = 0.5, what is cosec A?

Answer: 2

cosec A = 1/0.5 = 2.

Which two ratios multiply to give 1?

• sec A. sec A is hypotenuse / adjacent. It is the flip of cos A.
• cosec A — correct. Yes! cosec A = 1/sin A = hypotenuse / opposite.
• cot A. cot A is adjacent / opposite. It is the flip of tan A.
• sin A and cos A. sin A × cos A is not 1 for the 3-4-5 triangle: 3/5 × 4/5 = 12/25.
• sin A and sec A. sin A goes with cosec A. sec A is the flip of cos A.
• cos A and sec A — correct. Yes! sec A = 1/cos A, so cos A × sec A = 1.