Problem 5 - Algebra and Trigonometry

Question

Find the exact values of \(\sin ^{-1} \frac{1}{2}\) and \(\tan ^{-1} 1 .\) Express your answer in degrees.

Step-by-Step Solution

Answer

The exact values are \( \text{sin}^{-1} \frac{1}{2} = 30^\text{°} \) and \(\tan^{-1} 1 = 45^\text{°} \).

Step 1: Understanding \(\text{sin}^{-1}\frac{1}{2}\)

The expression \(\text{sin}^{-1}\) represents the inverse sine function, which means we are looking for an angle whose sine is \(\frac{1}{2}\).

Step 2: Recognizing Common Sine Values

Recall that \(\frac{1}{2}\) is a well-known sine value. The angle whose sine is \(\frac{1}{2}\) is \(30^\text{°}\), since \(\text{sin} \(30^\text{°}\) = \frac{1}{2}\).

Step 3: Determining the Angle for \( \tan^{-1} 1 \)

The expression \( \tan^{-1} 1 \) represents the inverse tangent function, which means we are looking for an angle whose tangent is 1.

Step 4: Recognizing Common Tangent Values

Recall that the tangent of \45^\text{°}\ is 1, since \(\text{tan} \(45^\text{°}\) = 1\).

Step 5: Conclusion

From the previous steps, we find that \( \text{sin}^{-1} \frac{1}{2} = 30^\text{°} \) and \(\tan^{-1} 1 = 45^\text{°} \).

Key Concepts

Inverse Sine

The inverse sine function, written as \(\text{sin}^{-1}\), is used to determine the angle whose sine value is given. When we see \(\text{sin}^{-1} \frac{1}{2}\), it implies that we need to find an angle \(\theta \) such that \(\text{sin}(\theta) = \frac{1}{2}\). By recalling the unit circle and common angle values, we know that \(\text{sin}(30^\text{°}) = \frac{1}{2}\). Therefore, \(\text{sin}^{-1} \frac{1}{2} = 30^\text{°}\).

• The range of \(\text{sin}^{-1}\) is \(-90^\text{°} \text{ to } 90^\text{°}\) or \(-\frac{\pi}{2} \text{ to } \frac{\pi}{2}\) in radians.
• This function helps in solving problems where we need to find an angle with a known sine value.

Understanding inverse sine is crucial for solving various trigonometric problems.

Inverse Tangent

The inverse tangent function, denoted as \(\text{tan}^{-1}\), is used to find an angle whose tangent value is provided. For the expression \(\text{tan}^{-1} 1\), we seek an angle \(\theta\) such that \(\text{tan}(\theta) = 1\). When we think about the unit circle or common tangent values, we recognize that \(\text{tan}(45^\text{°}) = 1\). Thus, \(\text{tan}^{-1} 1 = 45^\text{°}\).

• The range of \(\text{tan}^{-1}\) is \(-90^\text{°} \text{ to } 90^\text{°}\) or \(-\frac{\pi}{2} \text{ to } \frac{\pi}{2}\) in radians.
• This function is particularly useful in scenarios where we need to determine an angle from its tangent value.

Mastering inverse tangent helps in comprehending more complex trigonometric relationships.

Common Angle Values

Common angle values are essential when working with trigonometric functions. These values help in quickly identifying the results of inverse trigonometric functions without needing a calculator. Here are a few common angle values and their corresponding sine and tangent values:

• For \(30^\text{°}, \text{sin} (30^\text{°}) = \frac{1}{2} \text{ and } \text{tan} (30^\text{°}) = \frac{1}{\text{tan} (60^\text{°})} \).
• For \(45^\text{°}, \text{sin} (45^\text{°}) = \frac{\text{\textoverline{2}}}{2} \text{ and } \text{tan} (45^\text{°}) = 1 \).
• For \(60^\text{°}, \text{sin} (60^\text{°}) = \frac{\text{\textoverline{3}}}{2} \text{ and } \text{tan} (60^\text{°}) = \text{\textoverline{3}} \).

Understanding and memorizing these angles and their trigonometric values can simplify many problems in trigonometry. They form the basis for solving various inverse trigonometric functions.
Utilizing these common values effectively can enhance problem-solving efficiency.