Problem 68 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ -\frac{23 \pi}{4} $$

Step-by-Step Solution

Answer

Therefore, the sine, cosine, and tangent of \(-\frac{23 \pi}{4}\) without using a calculator are: \(\sin(-\frac{23 \pi}{4}) = -\frac{1}{\sqrt{2}}\), \(\cos(-\frac{23 \pi}{4}) = -\frac{1}{\sqrt{2}}\), and \(\tan(-\frac{23 \pi}{4}) = 1\).

Step 1: Reduce the angle

To find an equivalent angle on the unit circle, let's first reduce the original angle modulo \(2\pi\) (or \( \frac{8}{4}\pi \), since these are equal), since one full rotation around the unit circle measures \(2\pi\) radians or \(8/4\pi\) radians. That is: \[ -\frac{23 \pi}{4} = -5\pi + \frac{3 \pi}{4}\] The equivalent angle between \(0\) and \(2\pi\) is therefore \( \frac{3 \pi}{4}\).

Step 2: Evaluate the trigonometric functions

Now that we have found an equivalent angle on the unit circle, we can find the values of the sine, cosine, and tangent of the angle. It is known that for \( \frac{\pi}{4}\), \(\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\), \(\cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\), and \(\tan(\frac{\pi}{4}) = 1\). Now, considering that for \(\frac{3\pi}{4}\), the sine is also positive but the cosine is negative in the second quarter, we have: sine: \(-\sin(\frac{3 \pi}{4}) = -\frac{1}{\sqrt{2}}\), cosine: \( -\cos(\frac{3 \pi}{4}) = \frac{-1}{\sqrt{2}}\), tangent: \( \tan(\frac{3 \pi}{4}) = -\frac{\sin(\frac{3 \pi}{4})}{\cos(\frac{3 \pi}{4})} = 1\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit, centered at the origin (0, 0) on a coordinate plane. Imagine a point moving around the circumference of this circle; the position of this point can describe any angle in trigonometry.

Angles on the unit circle can be measured in degrees or radians. Radians are particularly useful because they provide a direct relationship between the length of an arc on the unit circle and the angle it subtends at the center. As the exercise above shows, sometimes we need to find an angle that corresponds to multiple rotations around the unit circle. This is done by finding an equivalent angle within a single 0 to 2π radian rotation, which simplifies the process of evaluating trigonometric functions.

Determining the trigonometric functions for angles on the unit circle often relies on knowing which quadrant the terminal side of the angle lies in, as each quadrant has different signs (positive or negative) for sine, cosine, and tangent values.

Radian Measure

A radian is another way to measure angles, based on the radius of a circle. One radian equals the angle created when we take the radius of a circle and wrap it along the circle's edge. For a unit circle, this means that the arc length is equal to the radius for an angle of one radian.

The importance of radian measure is evident in calculus and in simplifying the evaluation of trigonometric functions, as seen in the exercise. To convert an angle from degrees to radians, we use the fact that 180 degrees is equivalent to \(\pi\) radians. Therefore, to reduce the given angle \( -\frac{23 \pi}{4} \) to its smallest positive equivalent, we use modulo \(2\pi\) operations. This transformed angle falls within the familiar first 2π radians (or one complete revolution), making it easier to work with the usual trigonometric function values on the unit circle.

Sine Cosine Tangent

The sine, cosine, and tangent functions are three primary trigonometric functions, each providing a ratio of sides of a right triangle when given a specific angle. When considering these functions on the unit circle, the sine of an angle is the y-coordinate, the cosine is the x-coordinate, and the tangent is the ratio of the sine to the cosine.

For the exercise, with angle \(\frac{3\pi}{4}\), we are in the second quadrant where sine is positive and cosine is negative. This knowledge allows us to evaluate the sign of each function correctly based on the quadrant. Importantly, the exercise shows that some angles, such as \(\pi/4\) and \(3\pi/4\), result in the same absolute values for sine and cosine, but differ in sign. This is crucial to correctly identifying the signs of trigonometric functions depending on the angle's position relative to the quadrants on the unit circle.