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Chapter 8: Problem 80

Solve each equation on the interval \(0 \leq \theta<2 \pi\) \(4(1+\sin \theta)=\cos ^{2} \theta\)

Short Answer

Expert verified
The solutions are \(\theta = \frac{2\pi}{3}\) and \(\theta = \frac{4\pi}{3}\).

Step by step solution

01

- Use a Trigonometric Identity

Rewrite \(\cos(2 \theta)\) using the double-angle identity: \(\cos(2 \theta) = 2 \cos^2 \theta - 1\). Substituting this into the equation gives us: \(2 \cos^2 \theta - 1 + 5 \cos \theta + 3 = 0\)
02

- Combine Like Terms

Simplify the equation by combining like terms: \(2 \cos^2 \theta + 5 \cos \theta + 2 = 0\)
03

- Solve the Quadratic Equation

Solve the quadratic equation \(2 \cos^2 \theta + 5 \cos \theta + 2 = 0\) by factoring. Let \(x = \cos \theta\). The equation becomes \(2x^2 + 5x + 2 = 0\). Factor this to get \((2x + 1)(x + 2) = 0\).
04

- Find the Values of x

Set each factor equal to zero and solve for \x\: \(2x + 1 = 0\, \ x = -1/2\) and \(x + 2 = 0\, \ x = -2\). Since \cos \theta\ can only be between \(-1\) and \(1\), discard \(x = -2\). Thus, \ \cos \theta = -1/2\.
05

- Solve for \ \theta \

Find \ \theta\ such that \ \cos \theta = -1/2\ in the interval \([0, 2 \pi)\). The values are \(\theta = \frac{2\pi}{3}\) and \(\theta = \frac{4\pi}{3}\).

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Key Concepts

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

Double-Angle Identity
The double-angle identity is a useful tool in trigonometry. It expresses trigonometric functions of double angles in terms of single angles. For instance, the double-angle identity for cosine is given by: \( \cos(2 \theta) = 2\cos^2 \theta - 1 \).

In the given problem, we used this identity to rewrite \( \cos(2 \theta) \) as \( 2\cos^2 \theta - 1 \). This substitution simplifies the trigonometric equation and sets it up for solving as a quadratic equation.

Understanding and applying double-angle identities can significantly simplify solving trigonometric equations.
Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, typically in the form \( ax^2 + bx + c = 0 \). It can often be solved by factoring, completing the square, or using the quadratic formula.

In this exercise, after rewriting the trigonometric equation using the double-angle identity, we obtained a quadratic equation in terms of \( \cos \theta \): \( 2\cos^2 \theta + 5\cos \theta + 2 = 0 \).

Solving this quadratic equation allows us to find the possible values for \( \cos \theta \), which then helps in determining \( \theta \).
Factoring
Factoring is the process of breaking down a complex expression into simpler parts (factors) that, when multiplied, give the original expression.

For the quadratic equation \( 2x^2 + 5x + 2 = 0 \), we set \( x = \cos \theta \). This equation can be factored into \( (2x + 1)(x + 2) = 0 \).

By setting each factor equal to zero, we find the solutions for \( x \). In our problem:
  • \( 2x + 1 = 0 \) gives \( x = -1/2 \)
  • \( x + 2 = 0 \) gives \( x = -2 \)
Since \( \cos \theta \) must lie between -1 and 1, we discard \( x = -2 \). So, the relevant solution is \( \cos \theta = -1/2 \).
Cosine Function
The cosine function \( \cos \theta \) is one of the primary trigonometric functions. It is defined as the adjacent side over the hypotenuse in a right triangle. The function takes values between -1 and 1.

In this problem, once we determined that \( \cos \theta = -1/2 \), we needed to find the corresponding \( \theta \) values within the interval \( [0, 2\pi) \).

The cosine function equals -1/2 at specific angles within this interval:
  • \( \theta = \frac{2\pi}{3} \)
  • \( \theta = \frac{4\pi}{3} \)

This gives us the solutions where the original equation holds true.

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Most popular questions from this chapter

If \(\sin \theta>0\) and \(\cot \theta<0,\) name the quadrant in which the angle \(\theta\) lies.

Area under a Curve The area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=a\) and \(x=b\) is given by $$ \tan ^{-1} b-\tan ^{-1} a $$ (a) Find the exact area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=0\) and \(x=\sqrt{3}\). (b) Find the exact area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=-\frac{\sqrt{3}}{3}\) and \(x=1\)

Use the following discussion. The formula $$ D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right] $$ Approximate the number of hours of daylight for any location that is \(66^{\circ} 30^{\prime}\) north latitude for the following dates: (a) Summer solstice \(\left(i=23.5^{\circ}\right)\) (b) Vernal equinox \(\left(i=0^{\circ}\right)\) (c) July \(4\left(i=22^{\circ} 48^{\prime}\right)\) (d) Thanks to the symmetry of the orbital path of Earth around the Sun, the number of hours of daylight on the winter solstice may be found by computing the number of hours of daylight on the summer solstice and subtracting this result from 24 hours. Compute the number of hours of daylight for this location on the winter solstice. What do you conclude about daylight for a location at \(66^{\circ} 30^{\prime}\) north latitude?

Find the exact value of each expression. $$ \tan \left(\sin ^{-1} \frac{4}{5}+\cos ^{-1} 1\right) $$

Find the exact value of each expression. $$ \tan \left(\sin ^{-1} \frac{3}{5}+\frac{\pi}{6}\right) $$
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