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Chapter 6: Problem 106

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Solve \(2 \cos \left(A+40^{\circ}\right)=\sqrt{2}\) for all degree solutions. What is the sum of the first two positive solutions? a. \(100^{\circ}\) b. \(360^{\circ}\) c. \(440^{\circ}\) d. \(280^{\circ}\)

Short Answer

Expert verified
The sum of the first two positive solutions is \(280^{\circ}\).

Step by step solution

01

Simplify the Equation

Start by isolating the cosine function in the equation. Given is \(2 \cos (A + 40^{\circ}) = \sqrt{2}\). To simplify, divide both sides by 2: \(\cos (A + 40^{\circ}) = \frac{\sqrt{2}}{2}\).
02

Identify possible angles

The value \(\cos \theta = \frac{\sqrt{2}}{2}\) corresponds to angles \(\theta = 45^{\circ} + 360^{\circ}k\) and \(\theta = 315^{\circ} + 360^{\circ}k\) in degrees, where \(k\) is an integer.
03

Solve for A using first set of angles

Substitute \(A + 40^{\circ} = 45^{\circ} + 360^{\circ}k\). Solving for \(A\), we have:\[A = 45^{\circ} - 40^{\circ} + 360^{\circ}k = 5^{\circ} + 360^{\circ}k,\] which gives the first set of solutions for \(A\).
04

Solve for A using second set of angles

Substitute \(A + 40^{\circ} = 315^{\circ} + 360^{\circ}k\). Solving for \(A\), we have:\[A = 315^{\circ} - 40^{\circ} + 360^{\circ}k = 275^{\circ} + 360^{\circ}k,\] which gives the second set of solutions for \(A\).
05

Find the first two positive solutions

Using the solutions from steps 3 and 4, the first positive solution is \(A = 5^{\circ}\) and the second positive solution (from the second set) is \(A = 275^{\circ}\).
06

Sum of the first two positive solutions

Add the first two positive solutions: \(5^{\circ} + 275^{\circ} = 280^{\circ}\). This sum corresponds to one of the given options.

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Key Concepts

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

Cosine Function
The cosine function, one of the fundamental trigonometric functions, is crucial in solving many types of equations. It is often represented as \( \cos(\theta) \), where \( \theta \) is the angle measured in degrees or radians. The cosine function is periodic with a period of \( 360^{\circ} \) in degrees, meaning that the cosine of any angle \( \theta \) will repeat every \( 360^{\circ} \).
  • The basic form of the cosine function graph is a wave that starts at \( 1 \) (for \( \theta = 0 \)), decreases to \( -1 \), and returns to \( 1 \) over a \( 360^{\circ} \) interval.
  • Knowing the periodic nature of cosine allows us to find multiple solutions for angles, which is essential for solving equations where cosine equates to a specific value.
Understanding how to manipulate and solve equations involving the cosine function is vital for mastering trigonometry problems.
Angle Solutions
The concept of angle solutions in trigonometry arises when solving equations that involve trigonometric functions. In our problem, we find angles that satisfy an equation based on known cosine values.
  • Common cosine reference values include \( \cos \theta = \frac{\sqrt{2}}{2} \), which corresponds to angles such as \( 45^{\circ} \) and \( 315^{\circ} \).
  • Angles can be expressed in general solutions, incorporating a full period, \( 360^{\circ} \), as \( \theta = \theta_0 + 360^{\circ}k \), where \( k \) is any integer.
This allows us to find all possible angles that solve the equation, not just one instance, as shown when solving for \( A \) in the exercise.
Degree Solutions
In trigonometry, degree solutions signify the specific angle measures that satisfy a given equation, expressed in degrees. These solutions form sequences of angles derived from initial solutions by adding multiples of the periodic cycle.
For the problem \( \cos (A + 40^{\circ}) = \frac{\sqrt{2}}{2} \), solutions are found by setting:
  • \( A + 40^{\circ} = 45^{\circ} + 360^{\circ}k \) leading to \( A = 5^{\circ} + 360^{\circ}k \)
  • \( A + 40^{\circ} = 315^{\circ} + 360^{\circ}k \) leading to \( A = 275^{\circ} + 360^{\circ}k \)
With these expressions, you can identify all angles that will satisfy the equation for any integer \( k \). This approach helps determine series of angles for thorough solutions in degrees.
Problem Solving Steps
Solving trigonometric equations effectively requires a step-by-step approach. Here's how to tackle the given exercise:
  • **Step 1:** Isolate the trig function by dividing both sides, yielding \( \cos(A + 40^{\circ}) = \frac{\sqrt{2}}{2} \).
  • **Step 2:** Identify standard angles for the given cosine value: \( \theta = 45^{\circ} + 360^{\circ}k \) and \( \theta = 315^{\circ} + 360^{\circ}k \).
  • **Step 3:** Solve for \( A \) by substituting the identified angle equations and isolating \( A \).
  • **Step 4:** Derive your general solutions: \( A = 5^{\circ} + 360^{\circ}k \) and \( A = 275^{\circ} + 360^{\circ}k \).
  • **Step 5:** Sum the solutions: The first positive solutions \( 5^{\circ} \) and \( 275^{\circ} \) lead to a sum of \( 280^{\circ} \).
Following these problem-solving steps ensures clarity and accuracy while solving trigonometric equations.

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Most popular questions from this chapter

Solve for \(\theta\) if \(0^{\circ} \leq \theta<360^{\circ}\). $$ \sqrt{3} \sin \theta+\cos \theta=\sqrt{3} $$

For each equation, find all degree solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). If rounding is necessary, round to the nearest tenth of a degree. Use your graphing calculator to verify each solution graphically. $$ 16 \cos 2 \theta-18 \sin ^{2} \theta=0 $$

Solve each equation for \(x\) if \(0 \leq x<2 \pi\). Give your answers in radians using exact values only. $$ 2 \sin ^{2} x-\cos x-1=0 $$

Find all solutions in radians using exact values only. $$ \cos ^{3} 5 x=-1 $$

Geometry If central angle \(\theta\) cuts off a chord of length \(c\) in a circle of radius \(r\) (Figure 9), then the relationship between \(\theta, c\), and \(r\) is given by $$ 2 r \sin \frac{\theta}{2}=c $$ Find \(\theta\), if \(c=\sqrt{3} r\).
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