problem solving with trigonometry **Problem 5:**Find the radian measure of the largest angle of the triangle whose sides have lengths 8, 9, and 10.**Explanation:**To solve this problem, we need to use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. The law is expressed as:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]In this triangle, let the side with length 10 be $ c $, the largest side, and the opposite angle be $ C $. This makes $ a = 8 $ and $ b = 9 $.Plug these values into the Law of Cosines to find $ \cos(C) $:\[ 10^2 = 8^2 + 9^2 - 2 \cdot 8 \cdot 9 \cdot \cos(C) \]\[ 100 = 64 + 81 - 144 \cdot \cos(C) \]\[ 100 = 145 - 144 \cdot \cos(C) \]\[ 144 \cdot \cos(C) = 145 - 100 \]\[ 144 \cdot \cos(C) = 45 \]\[ \cos(C) = \frac{45}{144} \]\[ \cos(C) = \frac{5}{16} \]Now, find $ C $ by taking the arccosine of $\frac{5}{16}$:\[ C = \arccos\left(\frac{5}{16}\right) \]Finally, this calculation will yield the radian measure of the angle $ C $, the largest angle in the triangle.

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. Find the radian measure of the largest angle of the triangle whose sides have lengths 8, 9, and 10.

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

problem solving with trigonometry

**Problem 5:**

Find the radian measure of the largest angle of the triangle whose sides have lengths 8, 9, and 10.

**Explanation:**

To solve this problem, we need to use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. The law is expressed as:

\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

In this triangle, let the side with length 10 be $ c $, the largest side, and the opposite angle be $ C $. This makes $ a = 8 $ and $ b = 9 $.

Plug these values into the Law of Cosines to find $ \cos(C) $:

\[ 10^2 = 8^2 + 9^2 - 2 \cdot 8 \cdot 9 \cdot \cos(C) \]

\[ 100 = 64 + 81 - 144 \cdot \cos(C) \]

\[ 100 = 145 - 144 \cdot \cos(C) \]

\[ 144 \cdot \cos(C) = 145 - 100 \]

\[ 144 \cdot \cos(C) = 45 \]

\[ \cos(C) = \frac{45}{144} \]

\[ \cos(C) = \frac{5}{16} \]

Now, find $ C $ by taking the arccosine of $\frac{5}{16}$:

\[ C = \arccos\left(\frac{5}{16}\right) \]

Finally, this calculation will yield the radian measure of the angle $ C $, the largest angle in the triangle.

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