Quotients of quotients
Sin A and cos A both have the hypotenuse on the bottom. Divide one by the other. What is left?
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- A right triangle with its three sides.
- Sin A and cos A, from the same sides.
- Divide sin A by cos A.
- The hypotenuse cancels. What is left is tan A.
Tan from sin and cos
In a right triangle, sin A ÷ cos A leaves tan A. Here is why.
Sin A is opposite ÷ hypotenuse. Cos A is adjacent ÷ hypotenuse. Divide sin A by cos A. Dividing by a fraction means multiplying by its flip. So sin A ÷ cos A is (opposite ÷ hypotenuse) × (hypotenuse ÷ adjacent). The hypotenuse cancels. What is left is opposite ÷ adjacent. That is tan A.
Cot A is the flip of tan A. So cot A = cos A ÷ sin A.
Rule: tan A = sin A ÷ cos A. cot A = cos A ÷ sin A.
Check with a 5-12-13 triangle where A faces the side 5. Here sin A = 5/13 and cos A = 12/13. Then 5/13 ÷ 12/13 = 5/13 × 13/12 = 5/12. Opposite ÷ adjacent is also 5/12.
The same works at 30°. Here sin 30° = 1/2 and cos 30° = √3/2. So tan 30° = 1/2 ÷ √3/2 = 1/2 × 2/√3 = 1/√3.