Chapter 4.4, Problem 105E

To determine

To evaluate the six trigonometric functions.

Answer to Problem 105E

  

Explanation of Solution

Given information:

Write a short essay explaining to a classmate how to evaluate the six trigonometric functions of any angle $\theta$ in standard position. Include an explanation of reference angles and how to use them, the signs of the functions in each of the four quadrants, and the trigonometric values of common angles. Include figures or diagrams in your essay.

Calculation:

First, we will explain how to find six trigonometric functions of an angle $\theta$ .

If at the place of angle $\theta$ , we have given the terminal side of $\theta$ , then we can find trigonometric functions as follow:

  

If we need to find trigonometric functions of angle between $x$ -axis and line $y=-2x$

Then the coordinates of terminal axis $\left(x,y\right)$ such that $y=-2x$ .i.e. $\left(-2,4\right)$

So, we will take distance between $\left(0,0\right)$ and $\left(-2,4\right)$ as r, distance between $\left(0,0\right)$ and

  $\left(-2,4\right)$ as $x$ and distance between $\left(0,0\right)$ and $\left(0,4\right)$ as y. Then

  $r=\sqrt{x^2+y^2},\sin \theta =\frac{y}{r},\cos \theta =\frac{x}{r},\tan \theta =\frac{y}{x},\cot \theta =\frac{x}{y},\sec \theta =\frac{r}{x}and\csc \theta =\frac{r}{y}$

Now, we will explain to find the angle, if ratios are given. The signs of the trigonometry functions are different in all IV quadrants.

In $I$ quadrant all trigonometric functions are positive, in II quadrant $\sin \theta$ is positive while $\cos \theta$ and $\tan \theta$ are negative, in III quadrant $\tan \theta$ is positive while $\sin \theta$ and $\cos \theta$ will be negative and in IV quadrant , $\cos \theta$ is positive while $\sin \theta$ and $\tan \theta$ will be negative

  

So, we can say $\cos \theta =-\frac{1}{2}$ gives two values of $\theta$ in $\left[0,2\pi \right]$ . One value will be in II quadrant, other in III quadrant, because as shown in picture, $\cos \theta$ is negative in II and III quadrant.

Now, if we don’t want to deal with obtuse or reflex angle, then we can introduce reference angle of a given angle represented by $\theta \text{'}$ . The trigonometric functions give same value with given and reference angle but may differ in sign.

Reference angle: The smallest angle between the terminal side and horizontal angle is called reference angle.

To find value of $\sin \frac{4\pi }{3}$ . $\frac{4\pi }{3}$ is in III quadrant , and sin dives −ve value in III quadrant . so its reference angle will be $\frac{4\pi }{3}-\pi =\frac{\pi }{3}$ and $\sin \frac{4\pi }{3}=-\sin \frac{\pi }{3}$ .

Now, if we need to find the trigonometric functions of angles given in radians then we can use calculator. Moreover, we can convert degrees in radians by taking $\pi =3.14$ .

Hence, In $I$ quadrant all trigonometric functions are positive, in II quadrant $\sin \theta$ is positive while $\cos \theta$ and $\tan \theta$ are negative, in III quadrant $\tan \theta$ is positive while $\sin \theta$ and $\cos \theta$ will be negative and in IV quadrant , $\cos \theta$ is positive while $\sin \theta$ and $\tan \theta$ will be negative.