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Chapter 1: Problem 34

Find all solutions of the given equation. $$ \cos ^{2} x+5 \cos x=6 $$

Short Answer

Expert verified
\( x = 2k\pi \) for any integer \( k \).

Step by step solution

01

Rewrite the Equation

Rewrite the given trigonometric equation in terms of a quadratic equation. Let \( y = \cos x \). Then the equation \( \cos^{2} x + 5 \cos x = 6 \) becomes \( y^{2} + 5y = 6 \).
02

Form the Quadratic Equation

Move all terms to one side to form a standard quadratic equation: \( y^{2} + 5y - 6 = 0 \).
03

Solve the Quadratic Equation

Factor the quadratic equation: \( (y + 6)(y - 1) = 0 \). Therefore, the solutions for \( y \) are \( y = -6 \) and \( y = 1 \).
04

Check for Valid Trigonometric Solutions

Remember that \( y = \cos x \) must lie within the range \( -1 \leq \cos x \leq 1 \). Thus, \( y = -6 \) is not a valid solution. The only valid solution is \( y = 1 \).
05

Solve for \( x \)

If \( \cos x = 1 \), then \( x = 2k\pi \) for any integer \( k \).

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined.
These identities simplify complex trigonometric expressions and are key tools in solving trigonometric equations.
Examples: the Pythagorean identity, such as \(\text{sin}^2 x + \text{cos}^2 x = 1\), and angle sum identities, like \( \text{sin}(a + b) = \text{sin} a \text{cos} b + \text{cos} a \text{sin} b \). Knowing these helps break down complex problems into manageable parts.
Quadratic Equations
A quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \).
It is solved using factoring, completing the square, or the quadratic formula. It has the form \( y = \text{cos} x \) in trigonometric equations.
In our case, we converted the given trigonometric equation \( \text{cos}^2 x + 5 \text{cos} x = 6 \) to a quadratic form by substituting \( y = \text{cos} x \: y^2 + 5y - 6 = 0 \). This makes it easier to solve by factoring.
Here, we find \( (y+6)(y-1)=0 \: y = -6 \) or \( y = 1 \), then revert back to trigonometric terms.
Cosine Function
The cosine function \( \text{cos} x \) is crucial in trigonometry. Its value is the x-coordinate of a point on the unit circle.
The function is even, periodic, and bounded, meaning it repeats its values in regular intervals and stays within a fixed range.
Its range is \(-1 \leq \text{cos} x \ \leq 1\).
For our solution, given the quadratic \( y = \text{cos} x \, \(y \ = -6 and \ y = 1 \), the only valid replacement within the function's range is \ y = 1 y\).
This simplifies solving the equation even more.
Trigonometric Range Validation
Trigonometric range validation ensures that solutions from manipulated equations remain valid in original trigonometric contexts.
The principle range for \( \text{cos} x \) is from -1 to 1. Any solution outside that range is invalid.
For our equation turned quadratic, \{\text y }\}: \ y = \text -6 \ and \ y = \ 1. Since -6 does not fall within that range, it's an invalid solution, leaving \ y = 1.
This step refines the solutions of trigonometric equations and makes sure they are accurate and plausible in trigonometric functions contexts.

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Most popular questions from this chapter

Trec Volume. The following function approximates the proportion \(V\) of a tree's volume that lies below height \(h:\) $$\begin{aligned}V(h)=& \sin ^{p}\left(\frac{\pi h}{2}\right) \sin ^{q}\left(\frac{\pi \sqrt{h}}{2}\right) \\ & \times \sin ^{r}\left(\frac{\pi \sqrt[3]{h}}{2}\right) \sin ^{s}\left(\frac{\pi \sqrt[4]{h}}{2}\right)\end{aligned}$$ In this equation, the exponents \(p, q, r\), and \(s\) are constants and \(h\) is the proportion of the total height of the tree. For example, \(h=0\) corresponds to the bottom of the tree, \(h=1\) corresponds to the top, and \(h=1 / 2\) corresponds to halfway up the tree. \({ }^{18}\) a) Compute \(V(0)\) and \(V(1)\). b) Explain your answers to part (a) in terms of tree volume.

Use a calculator to find the values of the following trigonometric functions. \(\cot 56^{\circ}\)

Show that \(1+\tan ^{2} t=\sec ^{2} t\). (Hint: Begin with the Pythagorean Identity of Theorem 6 and divide both sides by \(\cos ^{2} t\).)

Sound Waves. The pitch of a sound wave is measured by its frequency. Humans can hear sounds in the range from 20 to \(20,000 \mathrm{~Hz}\), while dogs can hear sounds as high as \(40,000 \mathrm{~Hz}\). The loudness of the sound is determined by the amplitude. \({ }^{22}\). The note \(A\) below middle \(C\) on a piano generates a sound modeled by the function \(g(t)=4 \sin (440 \pi t)\), where \(t\) is in seconds. Find the frequency of \(A\) below middle \(C\).

Use a calculator to find the values of the following trigonometric functions. \(\tan 5^{\circ}\)
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