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In AABC, ZC is a right angle. Find the remaining side and angles. b=40, c = 41 (Round to the nearest tenth as needed.) a=

In AABC, ZC is a right angle. Find the remaining side and angles. b=40, c = 41 (Round to the nearest tenth as needed.) a=

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Topic: Solving Right Triangles #### Problem Statement: In △ABC, ∠C is a right angle. Find the remaining side and angles. Given: - \( b = 40 \) - \( c = 41 \) Calculate: - \( a \) (Round to the nearest tenth as needed.) #### Solution: To find the unknown side \( a \) of the right triangle where the hypotenuse \( c \) is 41, and one of the legs \( b \) is 40, we use the Pythagorean theorem. The Pythagorean theorem states: \[ a^2 + b^2 = c^2 \] Given: - \( b = 40 \) - \( c = 41 \) Substitute the known values into the equation: \[ a^2 + 40^2 = 41^2 \] \[ a^2 + 1600 = 1681 \] Subtract 1600 from both sides of the equation to solve for \( a^2 \): \[ a^2 = 1681 - 1600 \] \[ a^2 = 81 \] Taking the square root of both sides to solve for \( a \): \[ a = \sqrt{81} \] \[ a = 9.0 \] Therefore, the remaining side \( a \) is 9.0 (rounded to the nearest tenth, though it's exactly 9 in this case). To find the remaining angles, we can use trigonometric ratios once we have all sides: ### Angle Calculation: - **Angle A (∠A)**: \[ \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{9}{41} \approx 0.2195 \] \[ \angle A \approx \sin^{-1}(0.2195) \approx 12.68^\circ \] - **Angle B (∠B)**: Since the sum of angles in a triangle is \(180^\circ\) and we know two angles (one being \(90^\circ\)): \[ \angle B = 90^\circ - \angle A \] \[ \angle B \approx 90^\circ - 12.68^\circ \] \[ \angle B \approx 77.32
Transcribed Image Text:### Topic: Solving Right Triangles #### Problem Statement: In △ABC, ∠C is a right angle. Find the remaining side and angles. Given: - \( b = 40 \) - \( c = 41 \) Calculate: - \( a \) (Round to the nearest tenth as needed.) #### Solution: To find the unknown side \( a \) of the right triangle where the hypotenuse \( c \) is 41, and one of the legs \( b \) is 40, we use the Pythagorean theorem. The Pythagorean theorem states: \[ a^2 + b^2 = c^2 \] Given: - \( b = 40 \) - \( c = 41 \) Substitute the known values into the equation: \[ a^2 + 40^2 = 41^2 \] \[ a^2 + 1600 = 1681 \] Subtract 1600 from both sides of the equation to solve for \( a^2 \): \[ a^2 = 1681 - 1600 \] \[ a^2 = 81 \] Taking the square root of both sides to solve for \( a \): \[ a = \sqrt{81} \] \[ a = 9.0 \] Therefore, the remaining side \( a \) is 9.0 (rounded to the nearest tenth, though it's exactly 9 in this case). To find the remaining angles, we can use trigonometric ratios once we have all sides: ### Angle Calculation: - **Angle A (∠A)**: \[ \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{9}{41} \approx 0.2195 \] \[ \angle A \approx \sin^{-1}(0.2195) \approx 12.68^\circ \] - **Angle B (∠B)**: Since the sum of angles in a triangle is \(180^\circ\) and we know two angles (one being \(90^\circ\)): \[ \angle B = 90^\circ - \angle A \] \[ \angle B \approx 90^\circ - 12.68^\circ \] \[ \angle B \approx 77.32
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