A right triangle has side lengths AC=7inches,BC=24 inches, and AB=25inches What are the measures of the angles in triangle ABC? mangle Aapprox 46.2^circ ,mangle Bapprox 43.8^circ ,mangle Capprox 90^circ mangle Aapprox 73.0^circ ,mangle Bapprox 17.0^circ ,mangle Capprox 90^circ mangle Aapprox 73.7^circ ,mangle Bapprox 16.3^circ ,mangle Capprox 90^circ mangle Aapprox 74.4^circ ,mangle Bapprox 15.6^circ ,mangle Capprox 90^circ

To find the measures of the angles in triangle ABC, we can use the trigonometric ratios sine, cosine, and tangent.<br /><br />Step 1: Calculate the sine of angle A using the formula:<br />sin(A) = opposite/hypotenuse<br />sin(A) = BC/AB<br />sin(A) = 24/25<br />sin(A) ≈ 0.96<br /><br />Step 2: Calculate the cosine of angle A using the formula:<br />cos(A) = adjacent/hypotenuse<br />cos(A) = AC/AB<br />cos(A) = 7/25<br />cos(A) ≈ 0.28<br /><br />Step 3: Calculate the tangent of angle A using the formula:<br />tan(A) = opposite/adjacent<br />tan(A) = BC/AC<br />tan(A) = 24/7<br />tan(A) ≈ 3.43<br /><br />Step 4: Calculate the measure of angle A using the inverse tangent function:<br />A = tan^(-1)(tan(A))<br />A ≈ tan^(-1)(3.43)<br />A ≈ 73.7°<br /><br />Step 5: Calculate the measure of angle B using the fact that the sum of the angles in a triangle is 180°:<br />B = 180° - A - C<br />B = 180° - 73.7° - 90°<br />B ≈ 16.3°<br /><br />Step 6: Calculate the measure of angle C using the fact that the sum of the angles in a triangle is 180°:<br />C = 180° - A - B<br />C = 180° - 73.7° - 16.3°<br />C ≈ 90°<br /><br />Therefore, the measures of the angles in triangle ABC are:<br />$m\angle A\approx 73.7^{\circ },m\angle B\approx 16.3^{\circ },m\angle C\approx 90^{\circ }$