Answer:
Smaller acute angle measures 16°.
Step-by-step explanation:
Calculate each angle by using trigonometric functions.
Step 1: Draw the right triangle.
See attached picture. Hypotenuse is always across the 90° angle and for the leg (which is 7) doesn't matter which of the drawn legs you choose. The two acute angles are x and y.
Step 2: Calculate angles.
We know that:
AB = 7 (leg)
CB = 25 (hypotenuse)
Calculate x.
From the reference angle x AB is adjacent and CB hypotenuse.
Recall SOH CAH TOA - SineOppositeHypotenuse CosineAdjacentHypotenuse TangentOppositeAdjacent
We have adjacent and hypotenuse so let's use cosine.
[tex]\cos{(\text{angle})} = \frac{\text{adjacent}}{\text{hypotenuse}}[/tex]
[tex]\cos{(x)} = \frac{\text{AB}}{\text{CB}}[/tex]
[tex]\cos{(x)} = \frac{7}{25}[/tex]
To get the angle x use function [tex]\cos^{-1}[/tex] on calculator. (arccos)
[tex]x = \cos^{-1}{(\frac{7}{25})}[/tex]
[tex]x \approx 73.739795^\circ[/tex]
Rounded to nearest degree:
[tex]x = 74^\circ[/tex]
Calculate y.
From the reference angle x AB is opposite and CB hypotenuse, so we can use sine.
[tex]\sin{(\text{angle})} = \frac{\text{opposite}}{\text{hypotenuse}}[/tex]
[tex]\sin{(y)} = \frac{\text{AB}}{\text{CB}}[/tex]
[tex]\sin{(y)} = \frac{7}{25}[/tex]
To get the angle y use function [tex]\sin^{-1}[/tex] on calculator. (arcsin)
[tex]y = \sin^{-1}{(\frac{7}{25})}[/tex]
[tex]y \approx 16.26020^\circ[/tex]
Rounded to nearest degree:
[tex]y = 16^\circ[/tex]
Smaller acute angle measures 16°.
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You could also calculate only one angle and use the fact that the sum of the angles in triangle is 180°.
x + y + 90° = 180°
But if you choose this approach make sure to not round too much in between and do that only at the end. So if you calculated x use 73.7398° or more decimals (more is even more accurate and better) and round to the nearest degree only in the end.
