Problem 67 - Algebra 2

Question

Solve each trigonometric equation for \(\theta\) in terms of \(y\) Sample $$\begin{aligned} y &=2 \sin 3 \theta+4 \\ \sin 3 \theta &=\frac{y-4}{2} \\\ 3 \theta &=\sin ^{-1}\left(\frac{y-4}{2}\right) \\ \theta &=\frac{1}{3} \sin ^{-1}\left(\frac{y-4}{2}\right) \end{aligned}$$ $$ y=2 \sin \theta $$

Step-by-Step Solution

Answer

\(\theta = \sin^{-1}\left(\frac{y}{2}\right)\)

Step 1: Isolate the trigonometric function

Rearrange the equation to isolate the trigonometric function on one side of the equation. For the second equation, we get: \(\sin \theta = \frac{y}{2}\)

Step 2: Apply the inverse trigonometric function

To get the \(\theta\) on its own, we need to apply the inverse sine function to both sides of the equation. The result is: \(\theta = \sin^{-1}\left(\frac{y}{2}\right)\)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial for solving equations where we need to find an angle given a trigonometric value. For instance, if you have a sine value and need to determine the angle, the inverse sine function, denoted as \( \sin^{-1} \), will help. This function essentially reverses the sine process. It's important to grasp that the inverse sine function does not output angles in degrees by default but rather in radians, which are commonly used in higher mathematics.
To apply these functions:

• Make sure the sine value is between -1 and 1, as these are the limits of the sine function.
• Calculate the angle using inverse trigonometric functions.
• Remember, \( \sin^{-1}(x) \) provides angles principally between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) for the sine function.

Inverse trigonometric functions simplify problems by transforming trigonometric back to angular measures, enabling the resolution of such problems.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all possible angles that satisfy a given trigonometric expression. In our exercise, the goal was to express \( \theta \) in terms of \( y \).
Here are the main steps to tackle such equations:

• **Isolate the Trigonometric Function:** Start by rearranging the equation so that the trigonometric part is alone on one side. For instance, if you have \( y = 2 \sin \theta \), isolate to \( \sin \theta = \frac{y}{2} \).
• **Apply the Inverse Function:** Use an inverse trigonometric function to solve for the angle, \( \theta = \sin^{-1}(\frac{y}{2}) \). This step converts your equation from the trigonometric domain to the angle domain.

While solving, it's important to consider all solutions within the domain of the trigonometric function, since angles can repeat every cycle (e.g., every \(2\pi\) for sine and cosine). Always check if you need the principal value or multiple angles for contexts like physics or engineering.

Sin Function

The sine function is a fundamental trigonometric function often denoted as \( \sin \). It relates an angle in a right-angle triangle to the ratio of the length of the side opposite to the angle and the hypotenuse. Sine functions are periodic, possessing characteristics that repeat every \(2\pi\) radians or 360 degrees.
Key properties of the sine function include:

• **Range and Domain:** The sine function takes inputs, or arguments, in terms of an angle (usually in radians) and provides an output between -1 and 1.
• **Periodicity:** The function is periodic, meaning \( \sin(\theta) = \sin(\theta + 2n\pi) \) for any integer \( n \).

In practical applications, sine functions model wave-like phenomena, including sound and light waves. When solving equations which include the sine function, always keep these properties in mind for basic understanding and to avoid common pitfalls.