Note that the given side is adjacent to the given angle and what we are looking for is the hypotenuse. Therefore, we will use the cosine ratio.
cos θ = adjacent/hypotenuse
In our triangle, we have that θ = 45^(∘) and the length of the adjacent leg is 8. We want to find the length of the hypotenuse, y. Let's substitute the values.
cos θ = adjacent/hypotenuse
⇒
cos 45^(∘) = 8/y
Take a look at the triangle. Since two of the angle measures are 45 and 90, by the Triangle Sum Theorem, the measure of the remaining angle is also 45. Since it is a 45^(∘)-45^(∘)-90^(∘) special right triangle, we can immediately determine the value of cos 45^(∘) as sqrt(2)2. Let's then substitute that value and solve for y.
