A right triangle has side lengths $A C=7$ inches, $B C=24$ inches, and $AB =25$ inches.
What are the measures of the angles in triangle $A B C$?
$m \angle A \approx 46.2^{\circ}, m \angle B \approx 43.8^{\circ}, m \angle C \approx 90^{\circ}$
$m \angle A \approx 73.0^{\circ}, m \angle B=17.0^{\circ}, m \angle C \approx 90^{\circ}$
$m \angle A \approx 73.7^{\circ}, m \angle B=16.3^{\circ}, m \angle C=90^{\circ}$
$m \angle A \approx 74.4^{\circ}, m \angle B=15.6^{\circ}, m \angle C \approx 90^{\circ}$
Answer (1)
Use the tangent function to relate the sides of the right triangle to angles A and B: tan ( A ) = 7 24 and tan ( B ) = 24 7 .
Calculate angle A by taking the arctangent of 7 24 : A = arctan ( 7 24 ) ≈ 73. 7 ∘ .
Calculate angle B by taking the arctangent of 24 7 : B = arctan ( 24 7 ) ≈ 16. 3 ∘ .
State the measures of all angles: m ∠ A ≈ 73. 7 ∘ , m ∠ B = 16. 3 ∘ , m ∠ C = 9 0 ∘ .
m ∠ A ≈ 73. 7 ∘ , m ∠ B = 16. 3 ∘ , m ∠ C = 9 0 ∘
Explanation
Analyze the problem and given data We are given a right triangle A BC with side lengths A C = 7 inches, BC = 24 inches, and A B = 25 inches. We need to find the measures of angles A , B , and C . Since it's a right triangle, we know that m ∠ C = 9 0 ∘ . We can use trigonometric ratios to find the measures of angles A and B .
Calculate angle A To find the measure of angle A , we can use the tangent function, which relates the opposite side to the adjacent side. In this case, the side opposite to angle A is BC = 24 inches, and the side adjacent to angle A is A C = 7 inches. Therefore, we have: tan ( A ) = A C BC = 7 24 To find the measure of angle A , we take the arctangent (inverse tangent) of 7 24 :
A = arctan ( 7 24 ) Using a calculator, we find that A ≈ 73.7 4 ∘ .
Calculate angle B To find the measure of angle B , we can use the tangent function again. In this case, the side opposite to angle B is A C = 7 inches, and the side adjacent to angle B is BC = 24 inches. Therefore, we have: tan ( B ) = BC A C = 24 7 To find the measure of angle B , we take the arctangent (inverse tangent) of 24 7 :
B = arctan ( 24 7 ) Using a calculator, we find that B ≈ 16.2 6 ∘ .
State the measures of all angles We already know that m ∠ C = 9 0 ∘ because it is a right triangle. Therefore, the measures of the angles in triangle A BC are approximately: m ∠ A ≈ 73. 7 ∘ m ∠ B ≈ 16. 3 ∘ m ∠ C = 9 0 ∘
Final Answer Comparing our calculated values with the given options, we find that the correct answer is: m ∠ A ≈ 73. 7 ∘ , m ∠ B = 16. 3 ∘ , m ∠ C = 9 0 ∘
Examples
Understanding angles in right triangles is crucial in many real-world applications. For example, when constructing a ramp, knowing the angle of elevation is essential for safety and accessibility. If you want to build a ramp that rises 2 feet over a horizontal distance of 10 feet, you can use the arctangent function to calculate the angle: a rc t an ( 10 2 ) ≈ 11. 3 ∘ . This ensures the ramp meets the required slope for easy use. Similarly, in navigation and surveying, right triangle trigonometry is used to determine distances and directions.
