Question

If a leg and a hypotenuse are 7 and 25 respectively, find the smaller of the two acute angles to the nearest degree.

Answer 1

Smaller acute angle measures 16°.

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.

`\cos{(\text{angle})} = \frac{\text{adjacent}}{\text{hypotenuse}}`

`cos((x)) = \frac{\text{AB}}{\text{CB}}`

`cos((x)) = (7)/(25)`

To get the angle x use function
`\cos^(-1)` on calculator. (arccos)

`x = \cos^(-1){((7)/(25))}`

`x \approx 73.739795^\circ`

Rounded to nearest degree:

`x = 74^\circ`

Calculate y.

From the reference angle x AB is opposite and CB hypotenuse, so we can use sine.

`\sin{(\text{angle})} = \frac{\text{opposite}}{\text{hypotenuse}}`

`sin((y)) = \frac{\text{AB}}{\text{CB}}`

`sin((y)) = (7)/(25)`

To get the angle y use function
`\sin^(-1)` on calculator. (arcsin)

`y = \sin^(-1){((7)/(25))}`

`y \approx 16.26020^\circ`

Rounded to nearest degree:

`y = 16^\circ`

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.

Answer 2

16.26 deg

Explanation:

sin(theta)=7/25

theta = inverse sin(7/25)

theta = 16.26