Question

find ( mangle r ).
triangle with right angle at t, sides rt = 5.7, st = 6.9, vertices r, t, s
write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle r = square^circ )

Explanation:

Step1: Identify the sides relative to angle R

In right triangle \( RST \) with right angle at \( T \), the side adjacent to \( \angle R \) is \( RT = 5.7 \), and the side opposite to \( \angle R \) is \( ST = 6.9 \). We can use the tangent function, which is defined as \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \).

Step2: Calculate the tangent of angle R

So, \( \tan(m\angle R)=\frac{ST}{RT}=\frac{6.9}{5.7} \). Let's compute this value: \( \frac{6.9}{5.7}\approx1.2105 \).

Step3: Find the angle using arctangent

To find \( m\angle R \), we take the arctangent (inverse tangent) of \( 1.2105 \). So, \( m\angle R=\arctan(1.2105) \). Using a calculator, \( \arctan(1.2105)\approx50.5^\circ \) (rounded to the nearest tenth).

Answer:

\( 50.5 \)