The Triangle Sum Theorem allows us to find the measure of ∠ Z.
m∠ X_(64^(∘)) + m∠ Y^(90^(∘)) + m∠ Z = 180^(∘)
⇕
m∠ Z = 26^(∘)
Finally, we use the right-hand side equation given by the Law of Sines and substitute m∠ Y = 90^(∘), m∠ Z = 26^(∘), and XZ=17 to find the value of XY.
In consequence, we've solved the given triangle by using the Law of Sines.
b Notice that △ XYZ is a right triangle and we know an acute angle and the hypotenuse.
Because the triangle above is a right triangle, we can use the trigonometric ratios. For example, we can find YZ by using the sine ratio.
sin 64^(∘) = YZ/XZ ⇒
YZ = 17sin 64^(∘)
By using a calculator, we get that YZ = 17sin 64^(∘) = 15.3. Next, the cosine ratio gives us the following relation.
cos 64^(∘) = XY/XZ ⇒
XY = 17cos 64^(∘)
We use again a calculator and get that XY = cos 64^(∘) = 7.5. Finally, the Corollary to the Triangle Sum Theorem tells us that the acute angles of a right triangle are complementary.
m∠ X_(64^(∘)) + m∠ Z = 90^(∘)
⇕
m∠ Z = 26^(∘)
In conclusion, we could solve the triangle by using the trigonometric ratios.
