We want to find the measure of its acute angles. Let's find each angle separately.
Finding the Measure of ∠ β
Note that we are given the opposite side to the unknown angle and the hypotenuse. Therefore, to find the value of ∠ β we will use the sine ratio.
sin β = Length of leg opposite to∠ β/Length of hypotenuse
In our triangle, we have that the length of the opposite leg to ∠ β and hypotenuse are 4 and 7.
sin β = Opposite/hypotenuse=4/7
The sine of the angle is 47. Now, to isolate β we will use the inverse function of sin.
sin β=4/7 ⇔ β=sin ^(- 1)(4/7)
Let's use a calculator to find the value of sin ^(- 1)( 47). First, we will set our calculator into degree mode. To do so, push MODE, select Degree instead of Radian in the third row, and push ENTER. Next, we push 2ND followed by SIN, introduce the value 47, and press ENTER.
Therefore, the measure of ∠ β correct to the nearest degree is about 35^(∘).
Finding the Measure of ∠ α
To find the measure of ∠ α, we will use the interior angles theorem.
∠ α + ∠ β + 90^(∘) = 180^(∘)
Let's substitute 35^(∘) for ∠ β and solve the above equation for ∠ α to find its measure.
