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Chapter 9: Problem 30

Find all solutions of the given equation. $$4 \cos \theta+1=0$$

Short Answer

Expert verified
\(\theta = 104.48^\circ + 360^\circ \cdot n\) and \(\theta = 255.52^\circ + 360^\circ \cdot n\), where \(n\) is an integer.

Step by step solution

01

Isolate the Trigonometric Function

We start with the equation \( 4 \cos \theta + 1 = 0 \). To isolate \( \cos \theta \), subtract 1 from both sides of the equation, getting \( 4 \cos \theta = -1 \). Then divide both sides by 4, yielding \( \cos \theta = -\frac{1}{4} \).
02

Determine Principal Value of Theta

The next step is to find the principal value of \( \theta \) where \( \cos \theta = -\frac{1}{4} \). Use the inverse cosine function \( \theta = \cos^{-1}(-\frac{1}{4}) \). Calculate \( \cos^{-1}(-\frac{1}{4}) \) using a calculator to find the nearest angle, which approximately equals \( 104.48^\circ \).
03

Find General Solutions

The cosine function is periodic with a period of \( 360^\circ \) (or \( 2\pi \) radians). The cosine function is negative in the second and third quadrants. Therefore, the general solutions are \( \theta = 104.48^\circ + 360^\circ \cdot n \) and \( \theta = 360^\circ - 104.48^\circ + 360^\circ \cdot n \), where \( n \) is any integer.
04

Simplify and Conclude Solutions

Simplify the second expression from the previous step: \( 360^\circ - 104.48^\circ = 255.52^\circ \). Thus, the general solutions are \( \theta = 104.48^\circ + 360^\circ \cdot n \) and \( \theta = 255.52^\circ + 360^\circ \cdot n \), with \( n \) being any integer.

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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, represented as \( \cos \theta \), is a fundamental part of trigonometry. It describes the relationship between the angle \( \theta \) and the adjacent and hypotenuse sides of a right triangle. When utilized in the unit circle, the cosine value of an angle is the x-coordinate of the corresponding point on the circle.Cosine values range between -1 and 1. When analyzing angles on a graph:
  • The function starts at 1 when \( \theta = 0 \).
  • It decreases to 0 at \( 90^\circ \, or \ \frac{\pi}{2} \) radians.
  • It reaches -1 at \( 180^\circ \, or \ \pi \) radians.
  • The cycle completes back at 1 when \( 360^\circ \, or \ 2\pi \) radians.
This repetitive pattern is what makes cosine a periodic function. Understanding this behavior is crucial for solving trigonometric equations, where finding specific cosine values can lead us to discover the angles corresponding to those values.
Inverse Trigonometric Functions
Inverse trigonometric functions allow us to determine angles when the trigonometric ratios are known. They play a crucial role in solving equations where the variable is an angle, like finding \( \theta \) when \( \cos \theta = -\frac{1}{4} \).The inverse cosine function is denoted as \( \cos^{-1} \). It outputs an angle whose cosine is a given value. For example, if \( x = -\frac{1}{4} \, then \ \theta = \cos^{-1}(x) \) will give us an angle \( \theta \).A few important aspects to remember about \( \cos^{-1} \):
  • The principal value range for \( \cos^{-1} x \) is typically between \( 0 \) to \( 180^\circ \) (or \( \pi \) radians).
  • This range includes angles in the first and second quadrants, where cosine values transition from positive to negative.
When using inverse trigonometric functions, a calculator or a unit circle can help to find the angle for specific cosine values.
Periodic Functions
A periodic function is one that repeats its values in regular intervals or periods. The cosine function is a classic example of a periodic function due to its consistent repeating pattern every \( 360^\circ \, or \ 2\pi \) radians.Cosine's periodicity is integral for finding all solutions to equations, such as finding all \( \theta \) where \( \cos \theta = -\frac{1}{4} \). Since \( \cos \ heta \) repeats its values, the solutions are not limited to one angle but repeat every complete cycle:
  • For any solution \( \ heta = \ heta_0 \, where \ heta_0 \ is the principal value, other solutions can be described as \ \ heta = \ heta_0 + 360^\circ \cdot n \ (or \ \theta = \ heta_0 + 2\pi \cdot n\) where \ n \ is any integer.
  • Additional solutions in the cosine-negative quadrants can be derived similarly, further emphasizing the importance of understanding how periodic functions operate within different quadrants.
This cyclical nature is what allows us to find all possible angles \( \ heta \) that satisfy a given trigonometric equation, making periodicity a key property in trigonometry.

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Most popular questions from this chapter

Three vectors u, v, and w are given. (a) Find their scalar triple product \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$\mathbf{u}=\langle 3,0,-4\rangle, \quad \mathbf{v}=\langle 1,1,1\rangle, \quad \mathbf{w}=\langle 7,4,0\rangle$$

Find the vectors \(\mathbf{u}+\mathbf{v}\) \(\mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}.\) $$\mathbf{u}=\langle a, 2 b, 3 c\rangle, \quad \mathbf{v}=\langle- 4 a, b,-2 c\rangle$$

Three vectors u, v, and w are given. (a) Find their scalar triple product \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .\) (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$\mathbf{u}=\mathbf{i}-\mathbf{j}+\mathbf{k}, \quad \mathbf{v}=-\mathbf{j}+\mathbf{k}, \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+\mathbf{k}$$

Solve the given equation. $$3 \sin ^{2} \theta-7 \sin \theta+2=0$$

Express the given vector in terms of the unit vectors i, \(j\), and \(\mathbf{k}\). $$\langle 12,0,2\rangle$$
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