Question

find ( mangle r ).
write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle r=square^{circ} )

Explanation:

Step1: Use the sine function

In a right - triangle (here \(\triangle RSQ\) with \(\angle S = 90^{\circ}\)), the sine of an angle in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle R\), the opposite side to \(\angle R\) is \(SQ\) (let's assume we can use the Pythagorean theorem \(SQ=\sqrt{7^{2}-5^{2}}=\sqrt{49 - 25}=\sqrt{24}\)), but we can also use the sine function directly. The formula for \(\sin R=\frac{SQ}{RQ}\), and by the Pythagorean theorem \(SQ=\sqrt{RQ^{2}-RS^{2}}\), but another way is to use the inverse sine function. We know that \(\sin R=\frac{SQ}{RQ}\), and from the right - triangle \(\sin R=\frac{\sqrt{RQ^{2}-RS^{2}}}{RQ}\), or more simply, using the basic right - triangle trigonometry \(\sin R=\frac{\text{opposite}}{\text{hypotenuse}}\). Wait, actually, we can use the cosine function. \(\cos R=\frac{RS}{RQ}\) (adjacent over hypotenuse). Given \(RS = 5\) and \(RQ=7\). So \(\cos R=\frac{5}{7}\).

Step2: Use the inverse cosine function

We know that if \(\cos R=\frac{5}{7}\), then \(R=\cos^{-1}(\frac{5}{7})\). Using a calculator, \(R=\cos^{-1}(\frac{5}{7})\approx44.4^{\circ}\)

Answer:

\(44.4\)