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 [tex]sin \theta= \frac{a^2-b^2}{a^2+b^2} [/tex] , then:
opposite = a² - b²
hypotenuse = a² + b²
By the Pythagorean theorem:
[tex]adjacent = \sqrt{(a^2+b^2)^2-(a^2-b^2)^2} \\= \sqrt{(a^2+b^2+a^2-b^2)(a^2+b^2-(a^2-b^2))} \\= \sqrt{(2a^2)(a^2+b^2-a^2+b^2)} \\= \sqrt{2a^2*2b^2}\\= \sqrt{4a^2b^2} \\ =2ab [/tex]
So, the other trigonometric ratios:
[tex]cos \theta= \frac{adjacent }{hypotenuse }= \frac{2ab}{a^2+b^2} \\ \\ \\ tan \theta= \frac{opposite}{adjacent }= \frac{a^2-b^2}{2ab} [/tex]
