Question

If sin A = y, then express cos A and tan A is terms of y.

Answer 1

Given: sin A = y, where –1 ≤ y ≤ 1.

Step-wise calculation:

1. Use the Pythagorean identity:

sin²A + cos²A = 1

2. Substitute sin A = y:

y² + cos²A = 1

⇒ cos²A = 1 – y²

3. Take square root:

`cos A = ±sqrt(1 - y^2)` 

The sign (±) is chosen according to the quadrant of A (i.e., the sign of cos A).

4. Compute `tan A = sin A/cos A` 

= `y/(±sqrt(1 - y^2))` 

= `± y/sqrt(1 - y^2)`, defined when cos A ≠ 0 (equivalently y ≠ ±1).

`cos A = ±sqrt(1 - y^2)` with the sign set by the quadrant of A, `tan A = y/cos A = ± y/sqrt(1 - y^2)` undefined when y = ±1 because then cos A = 0.