Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find each value of \(\theta : \mathbf{a} .\) in degrees \(\mathbf{b} .\) in radians $$ \theta=\arcsin \left(-\frac{\sqrt{3}}{2}\right) $$

Step-by-Step Solution

Answer

\( \theta = \boxed{-60^\circ} \) in degrees and \( \theta = \boxed{-\frac{\pi}{3}} \) in radians.

Step 1: Understand the Function

The function involved in this problem is the inverse sine function, \( \arcsin \left(-\frac{\sqrt{3}}{2}\right) \). This function outputs the angle whose sine value is \(-\frac{\sqrt{3}}{2}\). The range of \( \arcsin \) is typically \([-\frac{\pi}{2}, \frac{\pi}{2}]\) for real numbers.

Step 2: Identify the Reference Angle

The sine function has a positive value of \( \frac{\sqrt{3}}{2} \) at reference angles of \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \). Since we're dealing with a negative sine value, the reference angle remains \( \frac{\pi}{3} \), but we must find the correct angle in the defined range.

Step 3: Determine 90 in Radians

In the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle that gives \(-\frac{\sqrt{3}}{2}\) is located in the fourth quadrant. This makes the angle \(-\frac{\pi}{3}\), since sine is negative in this quadrant.

Step 4: Convert Radians to Degrees

To convert radians to degrees, use the conversion \(180^\circ = \pi \). Thus, \(-\frac{\pi}{3} \text{ radians} = -\frac{180}{3}^\circ = -60^\circ \).

Key Concepts

Radians and Degrees Conversion

Radians and degrees are two different units for measuring angles. Understanding the conversion between radians and degrees is vital for solving many trigonometric problems. Here's an easy way to think about it: the circle has 360 degrees or \(2\pi\) radians in a complete rotation.
The basic conversion formula is:

• To convert radians to degrees: Multiply the radian measure by \(\frac{180}{\pi}\).
• To convert degrees to radians: Multiply the degree measure by \(\frac{\pi}{180}\).

This means that if you have an angle measure in radians, like \(-\frac{\pi}{3}\), you can convert it to degrees as follows: \[-\frac{\pi}{3} \text{ radians} \times \frac{180}{\pi} = -60^\circ\]
Similarly, to go from degrees to radians, use the reverse operation. It is essential to familiarize yourself with these conversions for smooth navigation between the two measurements.

Reference Angles

Reference angles play a crucial role in understanding trigonometric functions. A reference angle is the smallest angle that the terminal side of a given angle can make with the x-axis.
Here are some key points to remember about reference angles:

• The reference angle is always positive.
• It measures between 0 to \(\frac{\pi}{2}\) radians, or 0 to 90 degrees.
• For a negative angle like \(-\frac{\pi}{3}\), think of the corresponding positive reference angle \(\frac{\pi}{3}\) by only considering the size of the angle without the sign.

By ensuring familiarity with reference angles, you can better interpret the trigonometric function’s behavior, especially when dealing with angles in different quadrants.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are found everywhere in mathematics, from geometry to calculus. Understanding these functions is key to solving equations involving angles.
The sine function, for instance, is defined as the ratio of the opposite side to the hypotenuse in a right triangle for an acute angle. When we examine its inverse, often written as \(\arcsin\), it means finding the angle whose sine value corresponds to a given ratio.
For example, when you encounter \(\theta = \arcsin \left(-\frac{\sqrt{3}}{2}\right)\), you're searching for the angle whose sine equals \(-\frac{\sqrt{3}}{2}\). From the unit circle, you'll find that this relates to angles where the sine function is negative, specifically in the third and fourth quadrants within the range \([\frac{-\pi}{2}, \frac{\pi}{2}]\). This specific angle becomes \(-\frac{\pi}{3}\) radians.
Recognizing and understanding these trigonometric functions allow for accurate analysis and interpretation of various mathematical situations.