Problem 33

Question

Solve the equation for $x$. $\arcsin (3 x-\pi)=\frac{1}{2}$

Step-by-Step Solution

Verified

Answer

The solution for the equation is $x = \frac{\pi+0.4794}{3}$.

1Step 1 - Apply the Sine Function

To isolate $x$, you must first get rid of the $\arcsin$ function. This can be done by applying the $\sin$ function to both sides of the equation. This gives: $\sin\left( \arcsin\left( 3x - \pi \right) \right) = \sin\left(\frac{1}{2}\right)$. The $\sin$ and $\arcsin$ are inverse functions of each other, so, to the left side you have just $3x - \pi$, giving: $3x - \pi = \sin\left(\frac{1}{2}\right)$.

2Step 2 - Apply the sine of one half

This operation uses the fact that $\sin\left(\frac{1}{2}\right)$ is a known value, approximately equal to 0.4794. So, substitute this value into the equation, resulting in: $3x - \pi = 0.4794$.

3Step 3 - Solve for $x$

Now, simplify the equation to isolate $x$. First, add $\pi$ to both sides of the equation, getting: $3x = \pi + 0.4794$. And then, divide all terms by 3, getting the solution for $x$: $x = \frac{\pi+0.4794}{3}$.

Key Concepts

Inverse Trigonometric FunctionsSine Function ApplicationAlgebraic Manipulation

Inverse Trigonometric Functions

Understanding inverse trigonometric functions is essential for solving equations like $\arcsin (3 x-\pi)=\frac{1}{2}$. These functions are the inverses of the trigonometric functions, meaning that where the sine function $\sin$ finds the ratio of the lengths of certain sides in a right triangle given an angle, the inverse sine function $\arcsin$ does the opposite: it finds the angle given that ratio.
For instance, if you have $\sin(\theta) = a/b$, then $\theta = \arcsin(a/b)$, where $\theta$ is an angle in radians or degrees. These inverse functions, including $\arcsin$, $\arccos$, and $\arctan$, are vital when trying to retrieve an angle from a given trigonometric value.
When dealing with $\arcsin$ specifically, keep in mind its range is between $ -\frac{\pi}{2} $ and $\frac{\pi}{2} $, meaning it will always return an angle in this interval. This understanding helps you know which side of the unit circle your angle will be, and thereby better visualize your solutions.

Sine Function Application

The sine function has numerous applications, both in theory and real-world scenarios. In the context of our exercise, using the sine function helps to undo the $\arcsin$ and progress towards isolating $x$. When you see an equation like $\arcsin (3 x-\pi)$, you'll need to apply the sine function to both sides to remove the inverse operation.
Remember that $\sin(\arcsin(x)) = x$ and $\arcsin(\sin(x)) = x$, for values of $x$ within the appropriate domain. Additionally, knowing specific values of the sine function is beneficial, like $\sin(\frac{1}{2})$ being approximately 0.4794, as it allows for direct substitution and simplification.

Real-World Uses of the Sine Function

In Physics, it helps calculate components of force vectors.
In Engineering, it aids in understanding wave behaviors.
In Architecture, it's useful for determining angles in structures.

In the given problem, applying the sine function leads to a clearer path to solve for $x$. This step mirrors what happens in various scientific and engineering tasks, where applying trigonometric functions is necessary to get meaningful data from given information.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill that you will often employ when working through trigonometric equations. After using a trigonometric or inverse trigonometric function, you typically have an equation that requires algebraic skills to solve for the variable of interest.
For example, in the given exercise, once we apply the sine function and substitute known values, we have $3x - \pi = 0.4794$. Algebraic manipulation involves recognizing what operations will isolate the variable. This might be addition or subtraction (to remove constants from one side of an equation), or multiplication or division (to deal with coefficients).

Steps of Algebraic Manipulation in Solving $x$:

Add $\pi$ to both sides to get rid of the subtraction of $\pi$.
Divide everything by 3 to solve for $x$ by removing the coefficient.

Mastering these steps allows you to move from a trigonometric expression to an algebraic one, ultimately finding the value you need. In more complex scenarios, you might need to factor expressions or work with quadratic equations, but the core idea remains the same: manipulate the equation algebraically until the variable is isolated and can be solved for.

Problem 33