Problem 23 - Contemporary Precalculus

Question

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$-23 \pi / 6$$

Step-by-Step Solution

Answer

Question: Compute the exact value of sine, cosine, and tangent of \(- 23\pi/6\). Answer: The exact values of sine, cosine, and tangent of the given angle \(- 23\pi/6\) are: $$ \sin{\left(\frac{-23\pi}{6}\right)} = \frac{1}{2} $$ $$ \cos{\left(\frac{-23\pi}{6}\right)} = \frac{\sqrt{3}}{2} $$ $$ \tan{\left(\frac{-23\pi}{6}\right)} = \frac{1}{\sqrt{3}} $$

Step 1: Determine the equivalent angle

We are given the angle \(-23\pi/6\). To find the equivalent angle in the first rotation (between 0 and \(2\pi\)), we need to add integer multiples of \(2\pi\). Since we have a negative angle, let's add \(4\pi\): $$ \frac{-23\pi}{6} + 4\pi = \frac{-23\pi + 24\pi}{6} = \frac{\pi}{6} $$ So, the equivalent angle in the first rotation is \(\frac{\pi}{6}\).

Step 2: Compute the sine, cosine, and tangent of \(\pi/6\)

Now that we have the equivalent angle in the first rotation, let's find the sine, cosine, and tangent of \(\frac{\pi}{6}\): 1. Sine: By definition, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. For a \(30^\circ\) angle (equivalent to \(\frac{\pi}{6}\) radians), the side opposite the angle is equal to half the length of the hypotenuse. Using the Pythagorean theorem, we find that the length of the adjacent side is \( \sqrt{3}/2 \) times the hypotenuse. $$ \sin{\frac{\pi}{6}} = \frac{1}{2} $$ 2. Cosine: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. For the angle \(\frac{\pi}{6}\), the adjacent side is equal to \( \sqrt{3}/2 \) times the hypotenuse. $$ \cos{\frac{\pi}{6}} = \frac{\sqrt{3}}{2} $$ 3. Tangent: The tangent of an angle is the ratio of the sine to cosine. So, for the angle \(\frac{\pi}{6}\): $$ \tan{\frac{\pi}{6}} = \frac{\sin{\frac{\pi}{6}}}{\cos{\frac{\pi}{6}}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} $$

Step 3: State the final results

The exact values of sine, cosine, and tangent of the given angle \(- 23\pi/6\) are: $$ \sin{\left(\frac{-23\pi}{6}\right)} = \sin{\frac{\pi}{6}} = \frac{1}{2} $$ $$ \cos{\left(\frac{-23\pi}{6}\right)} = \cos{\frac{\pi}{6}} = \frac{\sqrt{3}}{2} $$ $$ \tan{\left(\frac{-23\pi}{6}\right)} = \tan{\frac{\pi}{6}} = \frac{1}{\sqrt{3}} $$

Key Concepts

Exact Trigonometric Values

In trigonometry, finding exact values of sine, cosine, and tangent is crucial for solving problems without a calculator. These exact values are often derived from well-known angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\), which correspond to the radians \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) respectively.
For \(\frac{\pi}{6}\) (equivalent to \(30^\circ\)), the exact trigonometric values are:

• \(\sin{\frac{\pi}{6}} = \frac{1}{2}\)
• \(\cos{\frac{\pi}{6}} = \frac{\sqrt{3}}{2}\)
• \(\tan{\frac{\pi}{6}} = \frac{1}{\sqrt{3}}\)

These values come from the properties of a 30-60-90 triangle, where the ratios of the sides are always 1: \(\sqrt{3}\): 2. Knowing these values by heart is incredibly helpful for tackling various trigonometric questions quickly.

Reference Angles

Reference angles are essential in trigonometry as they help find the trigonometric values of various angles by relating them to angles between \(0\) and \(\frac{\pi}{2}\) (or \(0^\circ\) and \(90^\circ\)).
For example, if an angle is given in a negative direction or significantly larger than \(2\pi\), we can simplify it by calculating its reference angle.

The reference angle for \(-23\pi/6\) was found by adding \(4\pi\) to get \(\pi/6\):

• This process ensures the reference angle is within the first rotation.
• It allows you to use familiar trigonometric values.

Using reference angles simplifies calculations and helps visualize angles on the unit circle, making the process more intuitive and manageable.

Unit Circle

The Unit Circle is a powerful tool in trigonometry, offering a geometric way to understand the relationships between angles and their sine, cosine, and tangent values.
The circle has a radius of 1 and is centered at the origin (0,0) of a coordinate plane.

Key points include:

• The angle is measured from the positive x-axis.
• The coordinates of a point on the unit circle are \((\cos{\theta}, \sin{\theta})\).
• For example, for \(\theta = \frac{\pi}{6}\), the coordinates are \((\frac{\sqrt{3}}{2}, \frac{1}{2})\).

Understanding the unit circle is fundamental in trigonometry, as it provides a visual representation and an intuitive understanding of how angles and trigonometric functions relate. It also helps seamlessly connect the concepts of reference angles and exact values to real-world applications.