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Chapter 9: Problem 27

Compute each angle of the given triangle. Where necessary, use a calculator and round to one decimal place. $$a=7, b=8, c=13$$

Short Answer

Expert verified
Angles are approximately 120掳, 33.6掳, and 26.4掳.

Step by step solution

01

Identify the Type of Problem

Given the sides of a triangle, we need to find the angles. This can be solved using the Law of Cosines.
02

Use the Law of Cosines for Angle A

The Law of Cosines states: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(A) \). Substitute \( a = 7 \), \( b = 8 \), and \( c = 13 \):\[ 13^2 = 7^2 + 8^2 - 2 \, \cdot \, 7 \, \cdot \, 8 \, \cdot \, \cos(A) \]\[ 169 = 49 + 64 - 112 \, \cos(A) \]\[ 112 \, \cos(A) = 113 - 169 \]\[ \cos(A) = \frac{-56}{112} \]Solve for \( A \):\[ A = \cos^{-1}\left(-0.5\right) \]Using a calculator, \( A \approx 120^{\circ} \).
03

Use the Law of Cosines for Angle B

Now find angle \( B \) using similarly the Law of Cosines:\[ b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \]\[ 8^2 = 7^2 + 13^2 - 2 \, \cdot \, 7 \, \cdot \, 13 \, \cdot \, \cos(B) \]Substitute the values:\[ 64 = 49 + 169 - 182 \, \cos(B) \]\[ 182 \, \cos(B) = 218 - 64 \]\[ \cos(B) = \frac{154}{182} \]Solve for \( B \):\[ B = \cos^{-1}\left(\frac{154}{182}\right) \]Using a calculator, \( B \approx 33.6^{\circ} \).
04

Use the Angle Sum Property for Angle C

Since the angles in a triangle sum to \(180^{\circ}\), calculate \(C\) as follows: \[ C = 180^{\circ} - A - B \]. So, \[ C = 180^{\circ} - 120^{\circ} - 33.6^{\circ} \approx 26.4^{\circ} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Angles
When dealing with triangles, understanding their angles is crucial. A triangle has three angles that always sum to 180 degrees. Each triangle can have a distinctive angle combination:
  • An acute triangle has all angles less than 90 degrees.
  • A right triangle has one angle exactly equal to 90 degrees.
  • An obtuse triangle has one angle greater than 90 degrees.
In our exercise, we are given the lengths of the sides of a triangle, and we need to find these angles. This is often done through geometric principles like the Law of Cosines, which helps us explore the relationships between the angles and sides.
Angle Calculation
Calculating angles using the Law of Cosines involves a methodical approach. The Law of Cosines relates the lengths of a triangle's sides to its angles, formulated as:\[c^2 = a^2 + b^2 - 2ab \cdot \cos(A)\]This equation can be rearranged to find an unknown angle if the side lengths are known.

Let's consider a scenario where we have side lengths, and we want to find angle \(A\):
  • Calculate \(c^2\) to begin plugging values into the equation.
  • Solve for \(\cos(A)\), rearranging the equation to isolate \(\cos(A)\).
  • Use an inverse cosine or arc cosine function, often on a calculator, to transform \(\cos(A)\) into angle \(A\).
Repeat similar steps for the other angles \(B\) and \(C\) by substituting corresponding side lengths, ensuring all calculations maintain dimensional consistency through proper squarings and operations.
Trigonometry
Trigonometry is the study of triangles and the relationships between their sides and angles. It's a foundational branch of mathematics used in various fields such as physics, engineering, and architecture.
In this exercise, we used trigonometric concepts to solve for the angles in a triangle using the sides given. Key elements include:
  • The Law of Cosines: This is central to finding angles when all three sides are known. It is particularly useful for non-right triangles, where other trigonometric ratios like sine or cosine cannot be directly applied without additional data.
  • Inverse Trigonometric Functions: These functions retrieve the angle from a given trigonometric ratio. For example, \(\cos^{-1}\) gives the angle whose cosine value matches the calculated ratio.
  • Angle Sum Property: Once two angles are determined, the third can always be found using the fact that the sum of angles in a triangle is 180掳.
Understanding these principles allows you to dissect and solve complex triangular shapes with confidence, highlighting the power and importance of trigonometry in problem-solving.

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Most popular questions from this chapter

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. $$x=3 t^{2}, y=2 t^{3}, \quad-2 \leq t \leq 2(\text {semicubical parabola})$$

(a) \(r \cos \theta=3\) (b) \(r \sin \theta=3\)

The polar equation of a line is given. In each case: (a) specify the perpendicular distance from the origin to the line; (b) determine the polar coordinates of the points on the line corresponding to \(\theta=0\) and \(\theta=\pi / 2 ;\) (c) specify the polar coordinates of the foot of the perpendicular from the origin to the line; (d) use the results in parts (a), (b), and (c) to sketch the line; and (e) find a rectangular form for the equation of the line. $$r \cos \left(\theta-\frac{\pi}{4}\right)=1$$

Express vector in terms of the unit vectors i and \(\mathbf{j}.\) $$\langle 4,-2\rangle$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=\cos t+t \sin t, y=\sin t-t \cos t(\text {involute of a circle})\) (a) \(-\pi \leq t \leq \pi\) (c) \(-4 \pi \leq t \leq 4 \pi\) (b) \(-2 \pi \leq t \leq 2 \pi\)
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