Question

determine the solution set of the following equation, where θ is an angle in degrees.
72 sin²θ + sin θ = 1

choose the correct solution set.
a. {173.6° + 360°n, 172.8° + 360°n, 186.4° + 360°n, 352.8° + 360°n, where n is any integer}
b. {6.4° + 360°n, 352.8° + 360°n, where n is any integer}
c. {6.4° + 360°n, 173.6° + 360°n, 187.2° + 360°n, 352.8° + 360°n, where n is any integer}
d. {173.6° + 360°n, 187.2° + 360°n, where n is any integer}
e. {172.8° + 360°n, 186.4° + 360°n, where n is any integer}

Explanation:

Step1: Rewrite the equation

Rewrite the given equation \(72\sin^{2}\theta+\sin\theta = 1\) as a quadratic equation in terms of \(\sin\theta\). Let \(x=\sin\theta\), then the equation becomes \(72x^{2}+x - 1=0\).

Step2: Solve the quadratic equation

Use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for the quadratic equation \(ax^{2}+bx + c = 0\). Here, \(a = 72\), \(b=1\), \(c=- 1\).

First, calculate the discriminant \(\Delta=b^{2}-4ac=(1)^{2}-4\times72\times(-1)=1 + 288=289\).

Then, \(x=\frac{-1\pm\sqrt{289}}{2\times72}=\frac{-1\pm17}{144}\).

We have two solutions for \(x\):

• When we take the plus sign: \(x=\frac{-1 + 17}{144}=\frac{16}{144}=\frac{1}{9}\approx0.111\)
• When we take the minus sign: \(x=\frac{-1-17}{144}=\frac{-18}{144}=-\frac{1}{8}=- 0.125\)

Step3: Find the angles for \(\sin\theta=\frac{1}{9}\)

We know that if \(\sin\theta=\sin\alpha\), then \(\theta=\alpha+360^{\circ}n\) or \(\theta = 180^{\circ}-\alpha+360^{\circ}n\), where \(n\in\mathbb{Z}\).

For \(\sin\theta=\frac{1}{9}\approx0.111\), \(\alpha=\arcsin(0.111)\approx6.4^{\circ}\). So the solutions are \(\theta = 6.4^{\circ}+360^{\circ}n\) and \(\theta=180^{\circ}- 6.4^{\circ}+360^{\circ}n=173.6^{\circ}+360^{\circ}n\)

Step4: Find the angles for \(\sin\theta=-\frac{1}{8}\)

For \(\sin\theta =-\frac{1}{8}=- 0.125\), \(\alpha=\arcsin(0.125)\approx7.2^{\circ}\). But since \(\sin\theta\) is negative, the angles are in the third and fourth quadrants.

In the third quadrant: \(\theta=180^{\circ}+7.2^{\circ}=187.2^{\circ}\) (so \(\theta = 187.2^{\circ}+360^{\circ}n\))

In the fourth quadrant: \(\theta=360^{\circ}-7.2^{\circ}=352.8^{\circ}\) (so \(\theta=352.8^{\circ}+360^{\circ}n\))

Step5: Combine all solutions

Combining the solutions from \(\sin\theta=\frac{1}{9}\) and \(\sin\theta=-\frac{1}{8}\), we get \(\theta = 6.4^{\circ}+360^{\circ}n\), \(\theta=173.6^{\circ}+360^{\circ}n\), \(\theta = 187.2^{\circ}+360^{\circ}n\), \(\theta=352.8^{\circ}+360^{\circ}n\), where \(n\) is any integer.

Answer:

C. \(\{6.4^{\circ}+ 360^{\circ}n,173.6^{\circ}+360^{\circ}n,187.2^{\circ}+360^{\circ}n,352.8^{\circ}+360^{\circ}n,\text{ where }n\text{ is any integer}\}\)