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Chapter 10: Problem 1

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=42 \mathrm{~cm}, \mathrm{BC}=37 \mathrm{~cm}, \mathrm{AC}=26 \mathrm{~cm}\)

Short Answer

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Question: Given a triangle ABC with side lengths AB = 42, BC = 37, and AC = 26. Find the interior angles at vertices A, B, and C. Answer: To find the interior angles at vertices A, B, and C, follow these steps: 1. Calculate angle A using the cosine rule. 2. Calculate angle B using the sine rule. 3. Calculate angle C using the angle sum theorem. 4. State the angles A, B, and C as the final solution.

Step by step solution

01

Calculate angle A using the cosine rule

Using the cosine rule, we can find angle A knowing the sides AB, BC, and AC. The formula for angle A is: \[A = \arccos{\frac{B^2 + C^2 - A^2}{2BC}}\] Substitute the values: \[A = \arccos{\frac{(37)^2 + (26)^2 - (42)^2}{2(37)(26)}}\] Evaluate the arccos expression to find the angle A in degrees.
02

Calculate angle B using the sine rule

Having found angle A, we can use the sine rule to find angle B. The sine rule states that: \[\frac{a}{\sin{A}} = \frac{b}{\sin{B}}\] Rearrange to find \(\sin{B}\): \[\sin{B} = \frac{b \sin{A}}{a}\] Substitute the values: \[\sin{B} = \frac{(37) \sin{A}}{(42)}\] Evaluate the expression to find \(\sin{B}\) and use the arcsine function to find angle B in degrees.
03

Calculate angle C using the angle sum theorem

Since the sum of the angles in a triangle equals 180 degrees, we can find angle C by subtracting both angles A and B from 180 degrees: \[C = 180^\circ - A - B\] Substitute the values of angle A and angle B to find angle C in degrees.
04

State the solution

We have successfully found the angles A, B, and C in the triangle ABC. The final solution is the triangle with vertices A, B, and C, their respective side lengths, and their respective angle measures.

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Most popular questions from this chapter

For questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=15 \mathrm{~cm}, \mathrm{AC}=23 \mathrm{~cm}, B=57^{\circ}\)

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