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Use the Law of Sines to solve the triangle. Round your answers to two decimal places. A = 60°, a = 6, c = 5 B = C = b =

Use the Law of Sines to solve the triangle. Round your answers to two decimal places. A = 60°, a = 6, c = 5 B = C = b =

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Additional Topics In Trigonometry
Section3.2: Law Of Cosines
Problem 64E
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**Using the Law of Sines** To solve this triangle using the Law of Sines, we need to find the unknown angles \( B \) and \( C \), and the unknown side \( b \). Round your answers to two decimal places. Given: - \( A = 60^\circ \) - \( a = 6 \) - \( c = 5 \) ### Law of Sines Formula The Law of Sines is given by: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] ### Step-by-Step Solution: 1. Since \( A = 60^\circ \) and \( a = 6 \), we can plug these values into the Law of Sines to find side \( b \) and angles \( B \) and \( C \). 2. First, find angle \( B \): \[ \frac{a}{\sin A} = \frac{c}{\sin C} \implies \frac{6}{\sin 60^\circ} = \frac{5}{\sin C} \] Solving for \( \sin C \): \[ \sin C = \frac{5 \sin 60^\circ}{6} \] \[ \sin 60^\circ = \frac{\sqrt{3}}{2} \] \[ \sin C = \frac{5 \cdot \frac{\sqrt{3}}{2}}{6} = \frac{5\sqrt{3}}{12} \] 3. Determine \( C \): \[ C = \arcsin\left(\frac{5\sqrt{3}}{12}\right) \] Using a calculator: \[ C \approx 39.23^\circ \] 4. To find angle \( B \): \[ B = 180^\circ - A - C = 180^\circ - 60^\circ - 39.23^\circ \] \[ B \approx 80.77^\circ \] 5. Finally, using the Law of Sines again to find side \( b \): \[ \frac{a}{\sin A} = \frac{
Transcribed Image Text:**Using the Law of Sines** To solve this triangle using the Law of Sines, we need to find the unknown angles \( B \) and \( C \), and the unknown side \( b \). Round your answers to two decimal places. Given: - \( A = 60^\circ \) - \( a = 6 \) - \( c = 5 \) ### Law of Sines Formula The Law of Sines is given by: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] ### Step-by-Step Solution: 1. Since \( A = 60^\circ \) and \( a = 6 \), we can plug these values into the Law of Sines to find side \( b \) and angles \( B \) and \( C \). 2. First, find angle \( B \): \[ \frac{a}{\sin A} = \frac{c}{\sin C} \implies \frac{6}{\sin 60^\circ} = \frac{5}{\sin C} \] Solving for \( \sin C \): \[ \sin C = \frac{5 \sin 60^\circ}{6} \] \[ \sin 60^\circ = \frac{\sqrt{3}}{2} \] \[ \sin C = \frac{5 \cdot \frac{\sqrt{3}}{2}}{6} = \frac{5\sqrt{3}}{12} \] 3. Determine \( C \): \[ C = \arcsin\left(\frac{5\sqrt{3}}{12}\right) \] Using a calculator: \[ C \approx 39.23^\circ \] 4. To find angle \( B \): \[ B = 180^\circ - A - C = 180^\circ - 60^\circ - 39.23^\circ \] \[ B \approx 80.77^\circ \] 5. Finally, using the Law of Sines again to find side \( b \): \[ \frac{a}{\sin A} = \frac{
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