Question

If sine of x equals square root of 2 over 2, what is cos(x) and tan(x)? Explain your steps in complete sentences. (10 points)

Answer 1

`\bf sin(\theta)=\cfrac{opposite}{hypotenuse} \qquad cos(\theta)=\cfrac{adjacent}{hypotenuse} \quad % tangent tan(\theta)=\cfrac{opposite}{adjacent}\\\\ -------------------------------\\\\`


`\bf sin(x)=\cfrac{√(2)}{2}\cfrac{\leftarrow opposite}{\leftarrow hypotenuse}\impliedby \textit{now, let's find the adjacent side} \\\\\\ \textit{using the pythagorean theorem}\\\\ c^2=a^2+b^2\implies \pm√(c^2-b^2)=a\qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \pm\sqrt{(2)^2-(√(2))^2}=a\implies \pm√(4-2)=a\implies \pm√(2)=a`

now, the adjacent side, or the x-coordinate for the angle, can be negative and positive, is positive in the I quadrant and negative in the II quadrant, with a positive y-coordinate... so it can be either, the +/- one.


`\bf cos(x)=\cfrac{\pm√(2)}{2}\qquad tan(x)=\cfrac{(√(2))/(2)}{\pm(√(2))/(2)}\implies tan(x)=\pm 1`