tan is sin divided by cos

The idea

tan A = sin A/cos A and cot A = cos A/sin A.

Two ratios, one new ratio

In the 3-4-5 triangle, sin A = 3/5 = 0.6 and cos A = 4/5 = 0.8. And tan A = 3/4 = 0.75.

Can you make 0.75 from 0.6 and 0.8?

Which one gives 0.75?

• 0.6 ÷ 0.8
• 0.6 × 0.8
• 0.6 + 0.8

Divide and cancel

Press the buttons: first divide sin A by cos A, then cancel what is the same on top and bottom. Then drag C to a new triangle and try again.

Why it works

tan A = sin A / cos A and cot A = cos A / sin A

Both sin A = BC/AC and cos A = AB/AC have AC underneath. When we divide, the AC cancels: tan A = (BC/AC) ÷ (AB/AC) = BC/AB. So tan A is sin A divided by cos A, and cot A is its flip, cos A divided by sin A.

Notes

tan A = sin A / cos A and cot A = cos A / sin A

Check yourself

sin A = 0.8 and cos A = 0.6. What is tan A? Give a decimal.

Answer: 1.33

tan A = 0.8 / 0.6 = 4/3, about 1.33.

Which is the same as cot A?

tan A = 0.75 and cos A = 0.8. What is sin A?

Answer: 0.6

sin A = tan A × cos A = 0.75 × 0.8 = 0.6.

Why does AC disappear when sin A is divided by cos A?

• sin A / cos A. That is tan A.
• cos A / sin A — correct. Yes! cot A = 1/tan A = cos A / sin A.
• 1 / cos A. That is sec A.
• AC is always 1.. AC can be any length. It is not always 1.
• Both fractions have AC underneath, so it cancels: (BC/AC) ÷ (AB/AC) = BC/AB. — correct. Yes! The same length under both fractions cancels, and BC/AB is tan A.
• AC is zero in a right triangle.. AC is the longest side. It is never zero.