Problem 95 - Algebra and Trigonometry

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, 2π) solving for \(7 \sin^2 x - 1 = 0\) can be calculated by finding \(x = \sin^{-1}\left(\sqrt{\frac{1}{7}}\right)\) and \(x = \sin^{-1}\left(-\sqrt{\frac{1}{7}}\right)\), correct to four decimal places, and using the periodic property of sine function to find all possible solutions.

Step 1: Solve for \(\sin^2 x\)

First, rearrange the equation \(7 \sin^2 x - 1 = 0\) to solve for \(\sin^2 x\) by adding 1 to each side of the equation and then dividing by 7 to isolate \(\sin^2 x\), resulting in \(\sin^2 x = \frac{1}{7}\).

Step 2: Find the value of sin x

Since \(\sin^2 x = \frac{1}{7}\), then \(\sin x = \sqrt{\frac{1}{7}}\) or \(\sin x = -\sqrt{\frac{1}{7}}\). The reason you have two potential solutions for \(\sin x\) is because \(\sin^2 x\) could be positive or negative. Remember to take the square root of both sides.

Step 3: Solve the trigonometric functions

Next, use the arcsin or sin-1 function on your calculator to find the possible values for x, correct to four decimal places: \(x = \sin^{-1}\left(\sqrt{\frac{1}{7}}\right)\) or \(x = \sin^{-1}\left(-\sqrt{\frac{1}{7}}\right)\). Each will give multiple possible solutions due to the periodic nature of the sine function.

Step 4: Find all the solutions in the interval [0, 2π]

In the given interval [0, 2π), you can add 2π to the solutions obtained above to get all possible solutions in the interval.

Key Concepts

Sinusoidal Functions

Sinusoidal functions are mathematical functions that simulate the behavior of waves. The sine function, represented as \( \sin(x) \), is one of these functions. It describes how a waveform oscillates between values, and in mathematical problems, it often appears squared as in \( \sin^2(x) \).
Understanding the nature of sinusoidal functions is crucial because:

• They repeat in cycles, known as periodicity.
• The function values oscillate between -1 and 1.
• They help model real-world phenomena such as sound and light waves.

The equation \( 7 \sin^2 x - 1 = 0 \) from the original exercise relies on this oscillatory behavior. Sinusoidal functions are naturally versatile, which is why they're common in many areas of mathematics and science.

Interval Notation

Interval notation is a concise way to express a range of values. In the context of solving trigonometric equations, it helps define where solutions are valid.
For instance, the interval \([0, 2\pi)\) specifies:

• The solutions start at 0 and go up to, but do not include, \(2\pi\).
• This interval covers one complete cycle of the sine function.
• It is crucial for determining all possible solutions within one period.

In the exercise, interval notation was used to find all valid solutions without repeating values.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \( \arcsin \) or \( \sin^{-1} \), are used to find angles when given a sine value.
Here's why they're important:

• They "undo" the sine action to retrieve the angle.
• In the exercise, \( \sin^{-1} \left( \sqrt{\frac{1}{7}} \right) \) is used to calculate the value of \( x \).
• These functions are bounded between specific intervals. For sine, it's \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

By employing the inverse function, you can systematically solve for the potential angles that meet the original equation conditions.

Calculator Usage

Using a calculator is vital for solving complex equations like the one given in the exercise.
When using a calculator to solve trigonometric equations:

• Ensure it's set to the correct mode, either degrees or radians. In this case, use radians.
• Enter values carefully to avoid rounding errors. The solutions need precision up to four decimal places.
• Utilize the \( \sin^{-1} \) function to compute the inverse sine values.

Proper calculator usage helps verify and confirm the solutions efficiently, ensuring that all answers fall within the specified interval.