Problem 18 - Algebra 2

Question

Solve each equation in the interval from 0 to 2\(\pi .\) Round to the nearest hundredth. \(-2 \cos \pi \theta=0.3\)

Step-by-Step Solution

Answer

The solution to the equation is \( \theta = 0.55 \)

Step 1: Isolate the Cosine Function

Divide both sides of the equation by -2 to isolate the cosine function. This gives you \( \cos \pi \theta = -\frac{0.3}{2} = -0.15 \).

Step 2: Apply Inverse Cosine to both sides

Apply the inverse cosine function to both sides of the equation to isolate \( \pi \theta \). This yields \( \pi \theta = \cos^{-1}(-0.15) \). Before moving forward, remember that the value of \( \cos^{-1}(-0.15) \) should be obtained in radians.

Step 3: Calculate Radian Value

Calculate \( \cos^{-1}(-0.15) \) in radians using a calculator. This will produce approximately a value of 1.72 radians.

Step 4: Isolate \( \theta \)

To find \( \theta \), divide both sides of the equation by \( \pi \). This gives \( \theta = \frac{1.72}{\pi} \). Calculate this value using a calculator. This gives approximately \( \theta = 0.55 \).

Step 5: Verify Interval

Check if the value of \( \theta \) obtained is within the required interval from 0 to 2\( \pi \). Since the value 0.55 lies within the interval from 0 to 2\( \pi \), the solution for \( \theta \) is valid.

Step 6: Round to the nearest hundredth

The final step is rounding the solution to the nearest hundredth. Here, it doesn't affect the solution, as the solution was already at the hundredth place. So, the final solution remains \( \theta = 0.55 \)

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, and it's essential to understanding how angles and triangles work. It specifically relates the angle in a right-angled triangle to the ratio of the adjacent side over the hypotenuse. The cosine function is crucial for solving trigonometric equations, such as \(-2 \cos \pi \theta=0.3\), by helping us isolate the variable we need to find.

• In a unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the angle's terminal side intersects the circle.
• The function is periodic, repeating every \(2\pi\) radians.
• The range of cosine values extends from -1 to 1, which defines the possible solutions for such equations.

When working with the cosine function, we're often tasked with finding angles given a cosine value, an operation that requires careful use of inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the tools we use to find angles when given a trigonometric ratio. In this context, the inverse cosine function, written as \(\cos^{-1}(x)\), is particularly useful. It allows us to determine an angle whose cosine is a specified value.

• The domain for inverse cosine is all real numbers between -1 and 1.
• The range for inverse cosine outputs is between 0 and \(\pi\) radians, because cosine is positive in the first quadrant and negative in the second quadrant.

In our exercise, applying the inverse cosine to \(-0.15\) gets us \(\pi \theta = \cos^{-1}(-0.15)\). This step is crucial for isolating and solving for the variable representing our angle in the equation.

Radian Measure

Radians are a unit of angular measure based on the radius of a circle. In trigonometry, using radians is often more beneficial than degrees. Here are a few things to know about radians:

• A full circle is \(2\pi\) radians, which is equivalent to 360 degrees.
• One radian is the angle made at the center of a circle by an arc whose length is equal to the circle's radius.
• Radians allow for easy integration and differentiation in calculus.

In our problem, once we have solved with the inverse cosine function in radians, we know we're asked to find solutions within \([0, 2\pi]\), the standard interval for a single rotation. This is a common requirement in trigonometry, ensuring results remain within realistic bounds for one complete circle.

Interval Notation

Interval notation is a mathematical notation used to represent subsets of real numbers. It is a concise way to define a range of numbers that includes all numbers lying between a set of endpoints.

• For example, the interval \([0, 2\pi]\) includes every point from 0 to \(2\pi\). This means the set of all numbers starting at 0, ending at \(2\pi\), and including both 0 and \(2\pi\).
• Brackets "[" and "]" means the boundary numbers are included.
• If parentheses "(" and ")" were used instead, the boundary numbers would not be included.

In trigonometry problems, such intervals help define the domain of solutions. They ensure that all calculations and final values lie within the boundaries necessary for the problem, such as the given rotational limits in trigonometric equations.