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B с a A b Note: Triangle may not be drawn to scale. Suppose ZA = 30° and c = 27. ZB = Find an exact value (report answer as a fraction, use sqrt if necessary): a =

B с a A b Note: Triangle may not be drawn to scale. Suppose ZA = 30° and c = 27. ZB = Find an exact value (report answer as a fraction, use sqrt if necessary): 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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### Trigonometry Problem Consider the right-angled triangle with vertices labeled as A, B, and C. The sides opposite to these vertices are labeled `a`, `b`, and `c` respectively. The right-angle is at vertex C. #### Given: - ∠A = 30° - Hypotenuse, c = 27 #### Problem 1: Calculate ∠B. #### Problem 2: Find the length of side a (express answer as a fraction and use `sqrt` if necessary). #### Diagram: In the diagram, the triangle is labeled with: - Hypotenuse `c`, opposite the right angle. - Side `a`, opposite vertex A. - Side `b`, opposite vertex B. The note mentions that the triangle may not be drawn to scale. ### Solutions: 1. **Angle ∠B:** Since the sum of angles in a triangle is 180°, and one of the angles is a right angle (90°): \[ \angle B = 90° - \angle A \] \[ \angle B = 90° - 30° \] \[ \angle B = 60° \] 2. **Length of side `a`:** Using the sine function in a right-angled triangle, \[ \sin(\angle A) = \frac{opposite}{hypotenuse} \] \[ \sin(30°) = \frac{a}{27} \] \[ \frac{1}{2} = \frac{a}{27} \] \[ a = 27 \cdot \frac{1}{2} \] \[ a = 13.5 \] Thus, the exact value of side `a` is: \[ a = \frac{27}{2} \] For educational purposes, ensure students understand the steps to solve for ∠B and side `a` using trigonometric identities and properties of triangles.
Transcribed Image Text:### Trigonometry Problem Consider the right-angled triangle with vertices labeled as A, B, and C. The sides opposite to these vertices are labeled `a`, `b`, and `c` respectively. The right-angle is at vertex C. #### Given: - ∠A = 30° - Hypotenuse, c = 27 #### Problem 1: Calculate ∠B. #### Problem 2: Find the length of side a (express answer as a fraction and use `sqrt` if necessary). #### Diagram: In the diagram, the triangle is labeled with: - Hypotenuse `c`, opposite the right angle. - Side `a`, opposite vertex A. - Side `b`, opposite vertex B. The note mentions that the triangle may not be drawn to scale. ### Solutions: 1. **Angle ∠B:** Since the sum of angles in a triangle is 180°, and one of the angles is a right angle (90°): \[ \angle B = 90° - \angle A \] \[ \angle B = 90° - 30° \] \[ \angle B = 60° \] 2. **Length of side `a`:** Using the sine function in a right-angled triangle, \[ \sin(\angle A) = \frac{opposite}{hypotenuse} \] \[ \sin(30°) = \frac{a}{27} \] \[ \frac{1}{2} = \frac{a}{27} \] \[ a = 27 \cdot \frac{1}{2} \] \[ a = 13.5 \] Thus, the exact value of side `a` is: \[ a = \frac{27}{2} \] For educational purposes, ensure students understand the steps to solve for ∠B and side `a` using trigonometric identities and properties of triangles.
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