Problem 95 - Precalculus Essentials

Question

Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$7 \sin ^{2} x-1=0$$

Step-by-Step Solution

Answer

The solutions for \(x\) in the interval \([0, 2\pi)\) that satisfy the equation \(7 \sin ^{2} x-1=0\) are \(x = \arcsin(\sqrt{1/7})\), \(x = \pi - \arcsin(\sqrt{1/7})\), \(x = \pi + \arcsin(\sqrt{1/7})\), \(x = 2\pi - \arcsin(\sqrt{1/7})\), \(x = \arcsin(-\sqrt{1/7})\), \(x = \pi - \arcsin(-\sqrt{1/7})\), \(x = \pi + \arcsin(-\sqrt{1/7})\), and \(x = 2\pi - \arcsin(-\sqrt{1/7})\), approximated to four decimal places.

Step 1: Rearrange the equation to isolate the sin^2(x) term

Add 1 to both sides of the equation, then divide by 7. This rearranges the equation to \(\sin ^{2} x = 1/7\).

Step 2: Solve for sin(x)

Take the square root of both sides to get \(\sin(x) = \pm \sqrt{1/7}\). Note that there are two solutions for \(\sin(x)\) - the positive and the negative square root.

Step 3: Solve for x

Use the arcsin function to solve for \(x\). \(\arcsin(\sqrt{1/7})\) and \(\arcsin(-\sqrt{1/7})\) give the two possible solutions for \(x\). \(x\) should be in the interval \([0, 2\pi)\), so \(x\) can have two possible values for each of the two arcsin values within the interval. Using a calculator, the following solutions for \(x\) are obtained: \(x = \arcsin(\sqrt{1/7})\), \(x = \pi - \arcsin(\sqrt{1/7})\), \(x = \pi + \arcsin(\sqrt{1/7})\), and \(x = 2\pi - \arcsin(\sqrt{1/7})\) for the positive square root, and \(x = \arcsin(-\sqrt{1/7})\), \(x = \pi - \arcsin(-\sqrt{1/7})\), \(x = \pi + \arcsin(-\sqrt{1/7})\), and \(x = 2\pi - \arcsin(-\sqrt{1/7})\) for the negative square root.

Key Concepts

Calculator Usage in Trigonometry

Navigating through trigonometric equations often involves the use of calculators, especially when the equations do not lend themselves to simple, exact solutions. A scientific calculator proves invaluable thanks to its ability to compute sine, cosine, tangent, and their inverse functions quickly.
In solving trigonometric equations, accuracy is key, and most calculators allow for results that are correct to several decimal places. For example, when faced with the equation given, you would first rearrange the terms algebraically before reaching for the calculator. Once you have the expression \(\sin(x) = \pm \sqrt{1/7}\), the calculator aids in computing the arcsine values and ensures your answers are accurate to four decimal places, as per the exercise requirements.

• Enter the square root and fraction into the calculator correctly to find \(\sin(x)\).
• Use the arcsin or inverse sine function for finding \(x\).
• Remember to check the calculator is in the correct mode (radians or degrees) based on the interval you're working within.

Always double-check each step in your calculations and verify your answers fall within the specified interval, here \( [0, 2\pi) \).

Sine Function Properties

The sine function is one of the fundamental trigonometric functions and displays a pattern which is periodic and oscillates between -1 and 1. Understanding the properties of the sine function is critical in solving equations like the one in our exercise.
Some key properties to be aware of are:

• The sine function has a period of \(2\pi\), which means it repeats its values every \(2\pi\) units.
• It is an odd function, meaning \(\sin(-x) = -\sin(x)\).
• The maximum value of the sine function is 1, and the minimum value is -1.

With these in mind, when solving for \(x\) in \(\sin^2(x) = 1/7\), one must consider all possible angles within the interval \( [0, 2\pi) \) whose sine squared gives you 1/7, while acknowledging the nature of the sine curve.

Arcsin Function Application

When you've isolated \(\sin(x)\) in an equation, as in our example, you will often apply the arcsin function to determine the angle \(x\). The arcsin function, known as the inverse sine, returns the angle whose sine is the given number.
This function is crucial because it allows us to work backward from the sine value to the angle. However, it's important to note that \(\arcsin(x)\) only returns the principal value, a single value usually in the range \( [-\frac{\pi}{2},\frac{\pi}{2}] \). This is why additional calculations are necessary to find all solutions within the specified interval of \( [0, 2\pi) \).

• The principal value \(\arcsin(\sqrt{1/7})\) gives the first solution for \(x\).
• Since the sine function is symmetric about the y-axis, a second solution is \(\pi - \arcsin(\sqrt{1/7})\).

You must then apply the periodic nature of sine to find any additional solutions in the given interval.

Trigonometric Identities and Equations

Trigonometric identities are equations that are true for all values of the variables involved. They can simplify the process of solving more complex trigonometric equations.
In the context of the given problem, understanding that \(\sin^2(x) + \cos^2(x) = 1\) can help. While it might not be directly applied to this particular equation, recognizing variations of this identity might aid in solving similar problems.

• Pythagorean identity: \(\sin^2(x) + \cos^2(x) = 1\)
• Co-function identities: These relate the trigonometric functions of complementary angles, such as \(\sin(\frac{\pi}{2} - x) = \cos(x)\) and vice versa.
• Even-odd identities: Reminds you that sine is an odd function, which helps in understanding its symmetry properties.

Leveraging these identities can greatly simplify the task of solving trigonometric equations and allow for a deeper understanding of the underlying relationships between trigonometric functions.