Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find each value of \(\theta : \mathbf{a} .\) in degrees \(\mathbf{b} .\) in radians $$ \theta=\arcsin \left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

\( \theta = -30^\circ \) or \( \theta = -\frac{\pi}{6} \).

Step 1: Understanding the Problem

We need to find the angle \( \theta \) for which the sine of the angle gives us \( -\frac{1}{2} \). We will find \( \theta \) in both degrees and radians.

Step 1: Finding the Principal Value in Degrees

The function \( \arcsin \) gives the principal value of the angle whose sine is the given number. For \( \arcsin \left( -\frac{1}{2} \right) \), the principal value in degrees is a common angle. We know from trigonometry that \( \sin(-30^\circ) = -\frac{1}{2} \). Therefore, the principal angle \( \theta \) in degrees is \( -30^\circ \).

Step 2: Finding the Principal Value in Radians

Convert the angle from degrees to radians using the conversion formula: \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180^\circ} \). Using \( \theta_{\text{degrees}} = -30^\circ \), we have:\[\theta_{\text{radians}} = -30 \times \frac{\pi}{180} = -\frac{\pi}{6}\]So, the principal value of \( \theta \) in radians is \( -\frac{\pi}{6} \).

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in trigonometry, involving the evaluation of angles in standard units. Angles can be measured in various ways, but the most common units are degrees and radians.
Degrees are perhaps the most familiar unit, where a full circle is divided into 360 equal parts. Therefore, an angle of 30 degrees represents a 30/360 fraction of a complete circle, which is measured from the center point.
Another way to express angle size is by using radians. A radian measures the angle created when the arc length equals the radius of the circle. Since the circumference of a circle is given by the formula \(2\pi R\), there are \(2\pi\) radians in a full circle. This means one complete rotation equals \(360^0\) or \(2\pi\) radians. Thus, 180 degrees is equivalent to \(\pi\) radians.

Degrees to Radians Conversion

Converting between degrees and radians is essential for solving various trigonometric problems. Knowing the conversion factor is crucial.
To convert degrees into radians, use the equation:
\[\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\]
This conversion formula stems from the relationship between the degree and radian measures, where \(180^0 = \pi\) radians.

• Example: To convert \(30^0\) to radians, multiply by \(\frac{\pi}{180}\): \(30 \times \frac{\pi}{180} = \frac{\pi}{6}\).
• Similarly, to convert radians back to degrees, use:
  \[\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}\]
• This method captures the precise mathematical relationship, ensuring accurate conversions for trigonometric applications.

Trigonometric Functions

Trigonometric functions are crucial in understanding relationships within right triangles. They relate the angles to the ratios of two sides of the triangle.
The six primary trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Each has its own role:

• Sine (\(\sin\)) represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\(\cos\)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)) equates to the opposite side divided by the adjacent side.
• Cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)) are the reciprocals of sine, cosine, and tangent respectively.

The inverse functions, like arcsine (\(\arcsin\)), are essential for finding angles given a ratio value. For example, \(\arcsin\left(-\frac{1}{2}\right)\) identifies the angle whose sine is \(-\frac{1}{2}\). Notably, \(\arcsin\)'s output range is restricted between \(-90^\circ\) and \(90^\circ\) (or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians), providing the principal angle measure.