The trigonometric function sinthetasinθ is equal to the ratio ("opposite side")/("hypotenuse")opposite sidehypotenuse. This is often shortened to sintheta=("opp")/("hyp")sinθ=opphyp.
To solve for the value of angle BAC∠BAC (also just called angle A∠A), we note that the side opposite of this angle is BC=5BC=5 and the hypotenuse is AB=5sqrt26AB=5√26. Our equation then becomes
sinA=("opp")/("hyp")=5/(5sqrt26)=1/sqrt26sinA=opphyp=55√26=1√26
So now we have
sinA=1/sqrt26sinA=1√26
and we want to solve for AA. This is done by applying the inverse function of sinsin (that is, sin^-1sin−1) to both sides:
sin^-1(sinA)=sin^-1(1/sqrt26)sin−1(sinA)=sin−1(1√26)
Applying sin^-1()sin−1() to sin()sin() undoes the sinsin function, so the LHS simply reduces to AA, and we get
A=sin^-1(1/sqrt26)approx11.31° (by calculator)
I'll leave angleB as practice for you; the answer is
Bapprox78.69°.
(Hint: there's a really easy shortcut.)
Remember: trig functions take in an angle and return a number between 0 and 1. Inverse trig functions thus take in a number between 0 and 1 and return an angle.
Side note: it is often sufficient to jump right from
sinA=b/c
to
A=sin^-1(b/c).
