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Exercises Trigonometry

Example 7 from Algebra and Trigonometry, 7.4 The Other Trigonometric Functions

Using Identities to Relate Trigonometric Functions

If $\cos (t) = \frac{12}{13}$ and $t$ is in quadrant IV, as shown in Figure 8, find the values of the other five trigonometric functions.

This is an image of graph of circle with angle of t inscribed. Point of (12/13, y) is at intersection of terminal side of angle and edge of circle.

Figure 8

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We can find the sine using the Pythagorean Identity, $\cos^{2} t + \sin^{2} t = 1,$ and the remaining functions by relating them to sine and cosine.

$\begin{matrix} {(\frac{12}{13})}^{2} + \sin^{2} t & = & 1 \\ \sin^{2} t & = & 1 - {(\frac{12}{13})}^{2} \\ \sin^{2} t & = & 1 - \frac{144}{169} \\ \sin^{2} t & = & \frac{25}{169} \\ sin t & = & \pm \sqrt{\frac{25}{169}} \\ sin t & = & \pm \frac{\sqrt{25}}{\sqrt{169}} \\ sin t & = & \pm \frac{5}{13} \end{matrix}$

The sign of the sine depends on the y-values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y-values are negative, its sine is negative, $- \frac{5}{13} .$

The remaining functions can be calculated using identities relating them to sine and cosine.

$\begin{matrix} tan t & = \frac{sin t}{cos t} & = \frac{- \frac{5}{13}}{\frac{12}{13}} & = - \frac{5}{12} \\ sec t & = \frac{1}{cos t} & = \frac{1}{\frac{12}{13}} & = \frac{13}{12} \\ csc t & = \frac{1}{sin t} & = \frac{1}{- \frac{5}{13}} & = \frac{- 13}{5} \\ cot t & = \frac{1}{tan t} & = \frac{1}{- \frac{5}{12}} & = - \frac{12}{5} \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

If $\sec (t) = - \frac{17}{8}$ and $0 < t < π,$ find the values of the other five functions.

How did it go?

PreviousUsing Identities to Simplify Trigonometric Expressions Next Finding the Values of Trigonometric Functions

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 7.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.