Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$3 \csc ^{2} \frac{x}{2}=2 \sec x$$

Step-by-Step Solution

Answer

The solutions for \( x \) in radians are \( x = \cos^{-1} \left( \frac{1}{4} \right) \) and \( x = 2\pi - \cos^{-1} \left( \frac{1}{4} \right) \). In degrees, \( \theta \approx 75.5^{\circ} \) and \( \theta \approx 284.5^{\circ} \).

Step 1: Understand Given Equation

The equation given is \( 3 \csc^2 \frac{x}{2} = 2 \sec x \).We need to find values of \( x \) (in radians) and \( \theta \) (in degrees) such that they satisfy this equation.

Step 2: Express in Terms of Sine and Cosine

Rewrite the cosecant and secant functions using sine and cosine:\( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).Substitute into the equation:\( 3 \left( \frac{1}{\sin \frac{x}{2}} \right)^2 = 2 \left( \frac{1}{\cos x} \right) \).

Step 3: Simplify and Solve for Trigonometric Functions

The equation becomes \( \frac{3}{\sin^2 \frac{x}{2}} = \frac{2}{\cos x} \).Multiply both sides by \( \sin^2 \frac{x}{2} \cos x \) to eliminate denominators:\( 3\cos x = 2\sin^2 \frac{x}{2} \).

Step 4: Use Double Angle Identity

Use the identity \( \sin^2 \frac{x}{2} = \frac{1 - \cos x}{2} \):Replace in the equation: \( 3\cos x = 2 \left( \frac{1 - \cos x}{2} \right) \).This simplifies to \( 3\cos x = 1 - \cos x \).

Step 5: Solve for \( \cos x \)

Rearrange equation to isolate \( \cos x \):\( 3\cos x + \cos x = 1 \) which simplifies to \( 4\cos x = 1 \).Thus, \( \cos x = \frac{1}{4} \).

Step 6: Find \( x \) in Radians

Use the inverse cosine function to find \( x \):\( x = \cos^{-1} \left( \frac{1}{4} \right) \).Since \( x \) can be in the first and fourth quadrants, solutions are:\( x = \cos^{-1} \left( \frac{1}{4} \right) \) and \( x = 2\pi - \cos^{-1} \left( \frac{1}{4} \right) \).

Step 7: Convert Radian Solution to Degrees

Convert solutions from radians to degrees using \( \theta = \frac{180}{\pi} \cdot x \):\( \theta = \cos^{-1} \left( \frac{1}{4} \right) \cdot \frac{180}{\pi} \).\( \theta = (360 - \cos^{-1} \left( \frac{1}{4} \right)) \cdot \frac{180}{\pi} \).

Key Concepts

Radians and Degrees

Understanding radians and degrees is crucial when working with trigonometric equations. These are the two primary units of measuring angles.

• A radian is the angle formed when the arc length is equal to the radius of a circle. It's a way of relating the angle directly to the circle.
• Degrees, on the other hand, break the circle into 360 equal parts. This unit is more intuitive and often used in day-to-day scenarios.

Converting between these units is simple:

• From degrees to radians: Multiply by \( \frac{\pi}{180} \)
• From radians to degrees: Multiply by \( \frac{180}{\pi} \)

In solving our equation, the answers were first found in radians and then converted to degrees.
Understanding this conversion helps relate abstract mathematical solutions to practical measurements.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find an angle when the value of a trigonometric function is known.

• The most common inverse functions are: \( \cos^{-1}(x) \), \( \sin^{-1}(x) \), and \( \tan^{-1}(x) \).
• These functions are essential for moving from a trigonometric ratio back to an angle.
• They return angle values measured in either degrees or radians.

In the given solution, \( \cos^{-1} \left( \frac{1}{4} \right) \) was used to find the angle whose cosine is \( \frac{1}{4} \).
This demonstrates how inverse trigonometric functions are pivotal in resolving trigonometric equations, converting between angle measures and ratios.

Double Angle Identity

The double-angle identities are useful formulas in trigonometry that expand or simplify expressions involving double angles. They take the form:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \)

These identities can transform expressions or solve equations where double angles appear.
In the exercise, we used the identity \( \sin^2 \frac{x}{2} = \frac{1 - \cos x}{2} \) to rewrite our equation involving a half-angle, allowing for further simplification.
This identity was crucial for simplifying the equation to the point where it could be easily solved, showcasing their utility in handling complex trigonometric forms.