First, let's remember the exact value of the sine and cosine of 45°. If we have a right triangle with sides 1 and hypotenuse sqrt(2), then we have:
Then, we have the following values for the sine and cosine of 45°:
[tex]\begin{gathered} \sin (45)=\frac{\text{opposite side}}{hypotenuse}=\frac{1}{\sqrt[]{2}} \\ \cos (45)=\frac{\text{adjacent side}}{hypotenuse}=\frac{1}{\sqrt[]{2}} \end{gathered}[/tex]
We can find the values of x and y using the trigonometric functions sine and cosine:
[tex]\begin{gathered} \sin (45)=\frac{\text{opposite side}}{hypotenuse} \\ \Rightarrow\sin (45)=\frac{x}{24} \\ \Rightarrow\frac{1}{\sqrt[]{2}}\cdot24=x \\ \Rightarrow x=(\frac{24}{\sqrt[]{2}})\cdot(\frac{\sqrt[]{2}}{\sqrt[]{2}})=\frac{24\cdot\sqrt[]{2}}{2}=12\sqrt[]{2} \\ \cos (45)=\frac{\text{adjacent side}}{hypotenuse} \\ \Rightarrow\cos (45)=\frac{y}{24} \\ \Rightarrow y=24\cdot(\frac{1}{\sqrt[]{2}})=12\sqrt[]{2} \end{gathered}[/tex]
therefore, the values for x and y are:
[tex]\begin{gathered} x=12\sqrt[]{2} \\ y=12\sqrt[]{2} \end{gathered}[/tex]
