Question

Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. $$\cos ^{-1}\left(-\frac{1}{2}\right)$$

Short answer

Answer: The value of the expression $$\cos ^{-1}\left(-\frac{1}{2}\right)$$ is $$150^\circ \text{ or } \frac{5\pi}{6} \text{ radians}$$

Step-by-step solution

Identify which special angles have cosine values of $$\pm\frac{1}{2}$$

For the angles $$30^\circ$$ (or $$\frac{\pi}{6}$$ in radians) and $$150^\circ$$ (or $$\frac{5\pi}{6}$$ in radians), their cosine values are positive $$\frac{1}{2}$$. However, the given value is $$-\frac{1}{2}$$, so we need to find the angles in Quadrants II and III where cosine has a negative value.

Determine the angles with cosine value of $$-\frac{1}{2}$$

In Quadrant II, the angle is $$180^\circ - 30^\circ = 150^\circ$$ (or $$\pi - \frac{\pi}{6} = \frac{5\pi}{6}$$ in radians). In Quadrant III, the angle is $$180^\circ + 30^\circ = 210^\circ$$ (or $$\pi + \frac{\pi}{6} = \frac{7\pi}{6}$$ in radians). Now we need to find which one of these angles is the correct answer for the inverse cosine function.

Understand the domain of inverse cosine function

The domain of the inverse cosine function is $$[0, \pi]$$ or $$[0^\circ, 180^\circ]$$, which means the output of the inverse cosine function falls within these angles.

Select the correct angle

Since the angle $$150^\circ$$ (or $$\frac{5\pi}{6}$$ in radians) falls within the domain of the inverse cosine function, it is the correct answer.

Write the final answer

The value of the given expression is: $$\cos ^{-1}\left(-\frac{1}{2}\right) = 150^\circ \text{ or } \frac{5\pi}{6} \text{ radians}$$

Key concepts

Inverse Cosine

The inverse cosine, denoted as \( \cos^{-1} \) or arccos, is one of the core functions in trigonometry that undoes the action of the cosine function. When you have a cosine value and you want to find the original angle that produced this value, you use the inverse cosine function. For example, if you know that the cosine of an angle is \(-\frac{1}{2}\), you can use the inverse cosine to find that angle.

It's essential to remember that the inverse cosine function will only provide angles that are within the domain of \([0, \pi]\) radians or \([0^\circ, 180^\circ]\). This is because the cosine function is not one-to-one over its entire range, so restrictions are necessary for its inverse to exist as a function. For the expression \(\cos^{-1}(-\frac{1}{2})\), we look for an angle in the specified domain where \(\cos\) is \(-\frac{1}{2}\). As learnt in the solution, this angle is \(150^\circ\) or \(\frac{5\pi}{6}\) radians.

Trigonometric Angle Values

Trigonometry revolves around the study of angles and their relationships to triangle sides, and the trigonometric angle values play a pivotal role in this field. The trigonometric functions, which include sine, cosine, and tangent, have specific values for special angles. These special angles often occur at multiples or divisions of \(30^\circ\), \(45^\circ\), and \(60^\circ\) or in radians as multiples of \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\).

For example, the cosine of \(30^\circ\) (or \(\frac{\pi}{6}\)) is \(\frac{\sqrt{3}}{2}\). Knowing these values is crucial for solving trigonometric equations and for evaluating the functions at these angles without a calculator. In the exercise, recognizing that the cosine has specific known values at special angles allowed for determining the result quickly.

Radians and Degrees

Radians and degrees are two units for measuring angles, and converting between them is a fundamental skill in trigonometry. One complete revolution around a circle is \(360^\circ\) in degrees or \(2\pi\) radians. To convert degrees to radians, we multiply by \(\frac{\pi}{180}\), and to convert radians to degrees, we multiply by \(\frac{180}{\pi}\).

For instance, if you wished to convert \(150^\circ\) to radians in the exercise solution, you would compute \(150^\circ \times \frac{\pi}{180}\), which equals \(\frac{5\pi}{6}\) radians. Understanding this conversion is crucial when working between systems that prefer radians (such as in calculus) and those that prefer degrees (such as in geometry).

Special Angles in Trigonometry

In trigonometry, special angles refer to angles whose sine, cosine, and tangent values are widely recognized and can be written in radical form or as simple fractions. These typically include angles such as \(30^\circ\), \(45^\circ\), \(60^\circ\), and their equivalences in radians. Knowledge of the trigonometric values for these angles can simplify many trigonometric problems since they allow for fast and precise calculation without using a calculator.

For instance, in our exercise, the cosine values for \(30^\circ\) and \(150^\circ\) (or \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\) radians respectively) are critical in identifying the angle associated with \(\cos^{-1}(-\frac{1}{2})\). Remembering the values and the quadrants in which each trigonometric function is positive or negative is also essential when working with these special angles.