Problem 1 - Precalculus Essentials

Question

Find the exact value of each expression. $$\sin ^{-1} \frac{1}{2}$$

Step-by-Step Solution

Answer

The exact value of \(\sin ^{-1} 1/2\) is 30 degrees or \(\pi / 6\) radians.

Step 1: Identify the standard angles

The sine function produces an output of 1/2 for two standard angles: 30 and 150 degrees. However, considering the range of the arcsin function, arcsin(x) where -1 ≤ x ≤ 1, is only defined for outputs between -90 and 90 degrees, inclusive. Thus the 150 degrees angle is out of the acceptable range.

Step 2: Apply the inverse sine function

Since \(\sin ^{-1} 1/2\) is asking for the angle whose sine is 1/2, we can use the identified standard angle from the previous step. Consequently, \(\sin ^{-1} 1/2 = 30\) degrees.

Step 3: Express answer in radians (optional)

If the context requires the result in radians rather than degrees, remember that 180 degrees equals π radians. To convert, establish the proportion: 30 degrees is to 180 degrees as x radians is to π radians. Solving this, it yields \(x = \pi / 6\). Hence, the radian measure for the angle is \(\pi / 6\).

Key Concepts

Understanding the Sine Function

The sine function is one of the basic trigonometric functions that relates an angle in a right triangle to the ratio of the length of the side opposite that angle to the hypotenuse. This function is key to understanding relationships in both mathematics and physics.

• It is usually represented as \( \sin(\theta) \), where \( \theta \) is the angle in question.
• In the context of the unit circle, the sine of an angle corresponds to the y-coordinate of the point on the circle associated with that angle.
• The function has a range from -1 to 1, as these are the maximum and minimum values for the sine of any angle.

This function is periodic, repeating every 360 degrees (or \( 2\pi \) radians), which means for every full circle you add to the angle, the sine value returns to the same number.
Key points to remember about the sine include that \( \sin(0) = 0 \), \( \sin(90^\circ) \) (or \( \pi/2 \) radians) = 1, and its reflective property: \( \sin(\theta) = \sin(180^\circ - \theta) \). These properties are useful when solving problems related to angles.

Exploring the Arcsin Function

The arcsin function, also known as the inverse sine function, is used to determine the angle whose sine is a given number. It is denoted as \( \sin^{-1} \) or \( \text{arcsin} \).

• It serves as the inverse operation of taking the sine of an angle. So if \( \sin(\theta) = x \), then \( \sin^{-1}(x) = \theta \).
• The domain of the arcsin function is restricted to values between -1 and 1, because these are the extents of values that the sine function can output.
• The range of arcsin is confined to -90 to 90 degrees (or \(-\pi/2 \) to \( \pi/2 \) radians), which ensures that the function gives a unique solution within this interval.

Understanding the arcsin function is crucial when dealing with trigonometric equations that require solving for angles. A common scenario might involve knowing the sine ratio and needing to find the corresponding angle, as we saw in the exercise with \( \sin^{-1}(1/2) = 30 \) degrees.

Angle Conversion Basics

Angle conversion is a useful mathematical skill that allows you to switch between different units of measuring angles, primarily between degrees and radians. Understanding this conversion helps in various fields such as trigonometry, calculus, and physics.

• Degrees and radians are two different units for measuring angles. There are 360 degrees in a full circle and \( 2\pi \) radians.
• The conversion factor between degrees and radians is \( \pi \) radians equals 180 degrees.
• To convert from degrees to radians, multiply the degree measurement by \( \pi/180 \).
• Conversely, to convert from radians to degrees, multiply the radian value by \( 180/\pi \).

For example, converting 30 degrees to radians involves setting up the proportion \( 30/180 = x/\pi \) which simplifies to \( x = \pi/6 \), thus confirming that 30 degrees is equivalent to \( \pi/6 \) radians. Mastery of these conversions is important for handling problems that require one specific unit of measurement.