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Chapter 6: Problem 23

In Exercises \(21-25,\) evaluate all six trigonometric finctions at the given number without using a calculator. $$\frac{7 \pi}{4}$$

Short Answer

Expert verified
Answer: The values of the six trigonometric functions for \(\frac{7\pi}{4}\) are \(\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \(\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\), \(\tan\left(\frac{7\pi}{4}\right) = -1\), \(\csc\left(\frac{7\pi}{4}\right) = -\sqrt{2}\), \(\sec\left(\frac{7\pi}{4}\right) = \sqrt{2}\), and \(\cot\left(\frac{7\pi}{4}\right) = -1\).

Step by step solution

01

Determine the Angle on the Unit Circle

Since we are given the angle \(\frac{7\pi}{4}\), we need to compute its measure on the unit circle. On the unit circle, various trigonometric functions are defined for angles which are multiples of \(30^\circ\), \(45^\circ\), and \(60^\circ\); note that these angles are equivalent to \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) in radians, respectively. Since our given angle \(\frac{7\pi}{4}\) is in radians, we can observe that it is \(7 \times \frac{\pi}{4} = \frac{7 \cdot 180^\circ}{4} = 315^\circ\). The similar angle for this angle in the first quadrant is the 45-degree reference angle, since \(360^\circ - 315^\circ = 45^\circ\).
02

Find the Unit Circle Coordinates

Now, we can find the coordinates of the point on the unit circle for the given angle. For a 45-degree reference angle in the fourth quadrant, the coordinates are \((x, y) = \left( \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2} \right)\).
03

Evaluate the Trigonometric Functions

Using the coordinates from Step 2, we can now evaluate all six trigonometric functions. 1. Sine: \(\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}\) 2. Cosine: \(\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\) 3. Tangent: \(\tan\left(\frac{7\pi}{4}\right) = \frac{\sin\left(\frac{7\pi}{4}\right)}{\cos\left(\frac{7\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1\) 4. Cosecant: \(\csc\left(\frac{7\pi}{4}\right) = \frac{1}{\sin\left(\frac{7\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\) 5. Secant: \(\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\cos\left(\frac{7\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\) 6. Cotangent: \(\cot\left(\frac{7\pi}{4}\right) = \frac{1}{\tan\left(\frac{7\pi}{4}\right)} = \frac{1}{-1} = -1\) So, the values of the six trigonometric functions for \(\frac{7\pi}{4}\) are \(\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \(\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\), \(\tan\left(\frac{7\pi}{4}\right) = -1\), \(\csc\left(\frac{7\pi}{4}\right) = -\sqrt{2}\), \(\sec\left(\frac{7\pi}{4}\right) = \sqrt{2}\), and \(\cot\left(\frac{7\pi}{4}\right) = -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
Imagine a circle with a radius of 1 unit centered at the origin of a coordinate system. This is the unit circle, a fundamental concept in trigonometry. It's a powerful tool for evaluating trigonometric functions because any point on its circumference corresponds to a set of trigonometric function values.

Each point on the unit circle can be defined by coordinates \( (x,y) \) that correspond to the cosine and sine of the angle formed by the radius line and the positive x-axis. Specifically, \( x = \cos(\theta) \) and \( y = \sin(\theta) \) where \( \theta \) is the angle in question. When dealing with angles greater than \( 360^\circ \) or \( 2\pi \) radians, the unit circle allows us to keep rotating and find equivalent angles that provide the same trigonometric values because of periodicity. In our example of \( \frac{7\pi}{4} \) radians, this angle corresponds to a full circle (\( 2\pi \) radians) minus \( \frac{\pi}{4} \) radians, landing us in the fourth quadrant.
Trigonometric Functions Without a Calculator
Mastering the calculation of trigonometric functions without a calculator is crucial for students. This skill hinges on understanding not just the unit circle but also the symmetries and periodic properties of trigonometric functions. By familiarizing themselves with key angles and their sine and cosine values, students can quickly determine trigonometric function values for any given angle.

Standard angles, such as \( 30^\circ, 45^\circ, \) and \( 60^\circ \), or in radian measure, \( \frac{\pi}{6}, \frac{\pi}{4}, \) and \( \frac{\pi}{3} \), serve as benchmarks. These specific angles have easily memorizable sine and cosine values, making it easier to compute the corresponding tangent, cotangent, secant, and cosecant values. Furthermore, knowing the sign of these functions in each quadrant aids in quickly determining the correct values.
Radian Measure
The radian measure is a way of expressing angles based on the radius of a circle. It is an alternative to degrees and is actually the preferred unit of angular measure in higher mathematics. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

In terms of conversion, \( 2\pi \) radians equal \( 360^\circ \), making it easy to switch between radians and degrees. For example, \( \frac{\pi}{4} \) radians is equivalent to \( 45^\circ \). When evaluating trigonometric functions, it's often simplest to work in radians, because the unit circle itself is radially measured. This is why, for instance, our given angle of \( \frac{7\pi}{4} \) is immediately understandable as a position on the unit circle once you are comfortable with radians.
Reference Angles
Reference angles are particularly useful when you're working with angles that are larger than \( 90^\circ \) (\( \frac{\pi}{2} \) radians). These are the acute angles formed by the terminal side of the angle and the x-axis. Regardless of which quadrant an angle ends up in, its reference angle will always be between \( 0 \) and \( \frac{\pi}{2} \) radians (or \( 0^\circ \) and \( 90^\circ \)) and can be used to find the values of trigonometric functions.

In practice, you find the reference angle for an angle in any quadrant by determining how far it is from the x-axis, either going clockwise or counterclockwise. In this case, the given angle \( \frac{7\pi}{4} \) radians has a reference angle of \( \frac{\pi}{4} \) radians since it is \( 2\pi - \frac{7\pi}{4} \) radians away from the positive x-axis. By knowing the trigonometric values of this reference angle in the first quadrant, where all are positive, we can easily determine the values for the given angle, considering the signs of functions in the fourth quadrant.

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Most popular questions from this chapter

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