Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\sin \theta=\frac{\sqrt{2}}{2}$$

Step-by-Step Solution

Answer

The acute angle \( \theta \) is \( \theta = \sin^{-1}\left( \frac{\sqrt{2}}{2} \right) = 45^\circ \).

Step 1: Recall the Special Angles

Recall the values for which the sine of an angle equals \( \frac{\sqrt{2}}{2} \). From trigonometry, we know that \( \sin(45^\circ) = \sin\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

Step 2: Use Inverse Sine Function

Use the inverse sine function to find \( \theta \) as an inverse trigonometric function. Recognize that \( \theta = \sin^{-1}\left( \frac{\sqrt{2}}{2} \right) \), which corresponds to the angle \( \frac{\pi}{4} \).

Step 3: Convert to Degrees

Convert \( \theta = \frac{\pi}{4} \) radians to degrees. Since \( 180^\circ = \pi \) radians, calculate \( \theta \) in degrees as follows: \( \theta = \frac{\pi}{4} \times \frac{180^\circ}{\pi} = 45^\circ \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in solving equations involving trigonometric ratios. They help us find the angle that corresponds to a given trigonometric value. In our exercise, we used the sine trigonometric function to identify an angle. The inverse sine function, often written as \( \sin^{-1}(x) \), helps find an angle whose sine is \( x \). Here, \( \theta = \sin^{-1}\left( \frac{\sqrt{2}}{2} \right) \).
Inverse trigonometric functions typically return an angle in a specific range:

• Inverse sine, \( \sin^{-1}(x) \), returns angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians or between \(-90^\circ\) and \(90^\circ\) degrees.
• Inverse cosine, \( \cos^{-1}(x) \), returns values between \(0\) and \(\pi\) radians or \(0^\circ\) to \(180^\circ\) degrees.
• Inverse tangent, \( \tan^{-1}(x) \), returns angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians.

These functions are reverse operations of the trigonometric functions, making them powerful tools for solving problems involving angles.

Radians to Degrees Conversion

To understand and work with angles, it's crucial to know how to convert between radians and degrees. Degrees and radians are both units for measuring angles, and being able to convert between the two allows seamless problem-solving in trigonometry and calculus.
Here's how the conversion works:

• The full circle in degrees is \(360^\circ\), while in radians, it is \(2\pi\).
• This means that \(180^\circ = \pi\) radians, which acts as our conversion factor.

This conversion allows us to switch back and forth:

• To convert from radians to degrees, multiply the radian measure by \(\frac{180^\circ}{\pi}\).
• To convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180^\circ}\).

In the exercise, our solution involved converting \(\frac{\pi}{4}\) radians into degrees, resulting in an angle of \(45^\circ\). Mastering this conversion process is essential for effective communication and problem-solving across various branches of mathematics.

Special Angles

Special angles are specific angles frequently encountered in trigonometry known for their easy-to-memorize sine, cosine, and tangent values. They include angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\).
For instance, memorizing that \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \) simplifies solving trigonometric equations. Such angles are often converted into radians, for example:

• \(45^\circ\) is \(\frac{\pi}{4}\) radians.
• \(30^\circ\) is \(\frac{\pi}{6}\) radians.
• \(60^\circ\) is \(\frac{\pi}{3}\) radians.

Recognizing these special angles and their trigonometric values helps quickly solve equations involving trigonometric functions. In the exercise, knowing that \(\sin(45^\circ)\) equals \(\frac{\sqrt{2}}{2}\) allows us to quickly find the acute angle \(\theta\) that satisfies the equation.