Problem 34 - 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. $$\cos x=\sin ^{2} \frac{x}{2}$$

Step-by-Step Solution

Answer

Solutions: \(x = 1.2305, 5.0527\) radians or \(x = 70.5^\circ, 289.5^\circ\).

Step 1: Expand the Equation

Recall that \sin^2 \frac{x}{2} = \frac{1 - \cos x}{2}\So we write the equation as: \[ \cos x = \frac{1 - \cos x}{2} \]

Step 2: Solve for \(\cos x\)

Multiply both sides by 2 to eliminate the fraction:\[ 2 \cos x = 1 - \cos x \] Add \(\cos x\) to both sides:\[ 3 \cos x = 1 \]

Step 3: Isolate Cosine

Solve for \(\cos x\):\[ \cos x = \frac{1}{3} \]

Step 4: Find General Solutions in Radians

Use the inverse cosine function, remembering that cosine is positive in the first and fourth quadrants:\[ x = \cos^{-1}(\frac{1}{3}) \approx 1.2305 \] To find the other solution, use symmetry in the circle:\[ x = 2\pi - 1.2305 = 5.0527 \]

Step 5: Convert Radians to Degrees and Round

Convert \(x \approx 1.2305\) radians and \(x \approx 5.0527\) radians to degrees:\[ x \approx 1.2305 \times \frac{180}{\pi} \approx 70.5^\circ \]\[ x \approx 5.0527 \times \frac{180}{\pi} \approx 289.5^\circ \]

Key Concepts

Radians and Degrees

When dealing with trigonometric equations, angle measurement is a critical aspect. Angles can be measured in both radians and degrees.
Although both units measure angles, they represent them differently on a circle.

• Degrees divide the circle into 360 parts, marking each part as one degree.
• Radians, on the other hand, use the radius of the circle for measurement. A full circle equates to exactly 2π radians.

Conversion between degrees and radians is key in trigonometry. The formula to convert degrees to radians is: \( ext{radians} = ext{degrees} \times \frac{\pi}{180}\).
Conversely, to go from radians to degrees, you use: \( ext{degrees} = ext{radians} \times \frac{180}{\pi}\).
This duality allows for flexibility depending on the context or preference in problem solving.

Cosine Function

The cosine function is one of the primary trigonometric functions. It measures the horizontal component of the unit circle. In simpler terms, it tells you how far along the x-axis a point is as you rotate around the circle.
The cosine of an angle is commonly denoted as \( ext{cos} \theta\). For practical calculations, cosine values range from -1 to 1.

• Cosine is positive in the first quadrant where both x and y values are positive.
• It becomes negative as you move towards the second quadrant.

Understanding where cosine is positive or negative helps in solving trigonometric equations. In the provided exercise, you solve for \( ext{cos} \theta = \frac{1}{3}\), indicating that the angle measure keeps cos in the positive domain.

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial for finding angles when given a trigonometric ratio. They perform the reverse operation of the normal trigonometric functions.
Instead of providing a ratio from an angle, they provide an angle from a ratio. The inverse cosine function, written as \( \cos^{-1}(x) \), is used in the given problem.

• This function identifies an angle whose cosine is a specific value.
• Due to cosine's periodic nature, inverse functions deliver principal values, typically between 0 and π for cosine.

Knowing how to use inverse trigonometric functions allows you to convert a ratio back into an angle measurement, solving parts of the equation neatly.

Angle Measures

Understanding angle measures in trigonometric equations is essential for interpreting results and ensuring coherence in problem-solving. Angles in trigonometry often relate to positions on the unit circle, allowing functions like sine and cosine to express relationships between angles and line segments.
In trigonometric equations, solving for angle measures involves ensuring that answers are the smallest nonnegative numbers possible. This means considering both radians and degrees, as applicable.

• Always utilize appropriate conversions when necessary for interpretation.
• Round off to specified precisions based on the requirements: radians often to four decimal places, degrees to the nearest tenth.

These practices ensure clarity and correctness in outcomes, as evidenced in the step-by-step problem-solving method demonstrated for this cosine equation.