Question

21. If sin Φ =a^2-b^2/a^2+b^2
find the values of other five trigonometric ratios.

Answer 1

Let’s label the sides of a right triangle a,b and c, and the angle opposite a as θ.

Then

sinθ=ac

cosθ=bc

tanθ=ab

and of course

a2+b2=c2

Usually you don’t have to write any of that out. We just fit that to our problem.

If we have a=2 and b=3 then tanθ=2/3 as required. So

c=a2+b2−−−−−−√=22+32−−−−−−√=4+9−−−−√=13−−√

sinθ=ac=213−−√=213−−√13

Note that c=−13−−√ also satisfies a2+b2=c2, so we’re not as sure about the sine as we like. All we really know is

sinθ=±213−−√13

For completeness, let’s also note

cosθ=bc=±313−−√=±313−−√13

We know since the tangent is positive the sine and cosine have the same sign, but we don’t really know if that’s plus or minus.

Check.

cos2θ+sin2θ=(±313−−√13)2+(±213−−√13)2=32⋅13132+2213132=(9+4)13132=1313132=1   ✓