Question

If sin θ = a² - b²/ a² + b² find the values of all the other trigonometric ratios

Answer 1

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse

If
`sin \theta= (a^2-b^2)/(a^2+b^2)` , then:
opposite = a² - b²
hypotenuse = a² + b²

By the Pythagorean theorem:

`adjacent = √((a^2+b^2)^2-(a^2-b^2)^2) \\= √((a^2+b^2+a^2-b^2)(a^2+b^2-(a^2-b^2))) \\= √((2a^2)(a^2+b^2-a^2+b^2)) \\= √(2a^2*2b^2)\\= √(4a^2b^2) \\ =2ab`

So, the other trigonometric ratios:


`cos \theta= (adjacent )/(hypotenuse )= (2ab)/(a^2+b^2) \\ \\ \\ tan \theta= (opposite)/(adjacent )= (a^2-b^2)/(2ab)`