A right triangle has side lengths [tex]$A C=7$[/tex] inches, [tex]$B C=24$[/tex] inches, and [tex]$AB =25$[/tex] inches.
What are the measures of the angles in triangle [tex]$A B C$[/tex]?
[tex]$m \angle A \approx 46.2^{\circ}, m \angle B \approx 43.8^{\circ}, m \angle C \approx 90^{\circ}$[/tex]
[tex]$m \angle A \approx 73.0^{\circ}, m \angle B \approx 17.0^{\circ}, m \angle C \approx 90^{\circ}$[/tex]
[tex]$m \angle A \approx 73.7^{\circ}, m \angle B \approx 16.3^{\circ}, m \angle C \approx 90^{\circ}$[/tex]
[tex]$m \angle A \approx 74.4^{\circ}, m \angle B \approx 15.6^{\circ}, m \angle C \approx 90^{\circ}$[/tex]
Answer (2)
Recognize that angle C is the right angle, so m ∠ C = 9 0 ∘ .
Use the sine function to find the measure of angle A: sin ( A ) = 25 24 , so m ∠ A ≈ 73. 7 ∘ .
Use the fact that the sum of the angles in a triangle is 180 degrees to find the measure of angle B: m ∠ B = 18 0 ∘ − m ∠ A − m ∠ C ≈ 16. 3 ∘ .
The measures of the angles are: m ∠ A ≈ 73. 7 ∘ , m ∠ B ≈ 16. 3 ∘ , m ∠ C = 9 0 ∘ .
Explanation
Problem Analysis We are given a right triangle A BC with side lengths A C = 7 inches, BC = 24 inches, and A B = 25 inches. Our goal is to find the measures of the angles in this triangle.
Identify the Right Angle Since A B is the longest side (25 inches), it is the hypotenuse. Therefore, angle C is the right angle, which means m ∠ C = 9 0 ∘ .
Calculate Angle A To find the measure of angle A , we can use the sine function, since we know the lengths of the opposite side ( BC ) and the hypotenuse ( A B ): sin ( A ) = A B BC = 25 24 Now, we calculate the measure of angle A by taking the inverse sine (arcsin) of 25 24 :
m ∠ A = arcsin ( 25 24 ) The result of this operation is approximately 73.7 4 ∘ .
Calculate Angle B To find the measure of angle B , we can use the fact that the sum of the angles in a triangle is 18 0 ∘ :
m ∠ B = 18 0 ∘ − m ∠ A − m ∠ C = 18 0 ∘ − 73.7 4 ∘ − 9 0 ∘ = 16.2 6 ∘ Alternatively, we can use the cosine function, since we know the lengths of the adjacent side ( A C ) and the hypotenuse ( A B ): cos ( B ) = A B A C = 25 7 Now, we calculate the measure of angle B by taking the inverse cosine (arccos) of 25 7 :
m ∠ B = arccos ( 25 7 ) The result of this operation is approximately 16.2 6 ∘ .
Final Answer Therefore, the measures of the angles in triangle A BC are approximately: 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. Similarly, in navigation, pilots and sailors use angles to determine their course and avoid obstacles. In architecture, calculating angles is vital for designing stable and aesthetically pleasing structures. These are just a few examples of how the principles of trigonometry and right triangles are applied in practical scenarios.
The measures of the angles in triangle ABC are approximately m ∠ A ≈ 73. 7 ∘ , m ∠ B ≈ 16. 3 ∘ , and m ∠ C = 9 0 ∘ .
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