7.4 The other trigonometric functions  (Page 4/14)

Alternate forms of the pythagorean identity

We can use these fundamental identities to derive alternate forms of the Pythagorean Identity, ${\cos }^{2}t+{\sin }^{2}t=1.$ One form is obtained by dividing both sides by ${\cos }^{2}t.$

The other form is obtained by dividing both sides by ${\sin }^{2}t.$

Using identities to relate trigonometric functions

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

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.

$$\begin{array}{ccc}\hfill {\left(\frac{12}{13}\right)}^2+{\sin }^{2}t& =& 1\hfill \\ \hfill {\sin }^{2}t& =& 1-{\left(\frac{12}{13}\right)}^2\hfill \\ \hfill {\sin }^{2}t& =& 1-\frac{144}{169}\hfill \\ \hfill {\sin }^{2}t& =& \frac{25}{169}\hfill \\ \hfill \text{sin }t& =& \pm \sqrt{\frac{25}{169}}\hfill \\ \hfill \text{sin }t& =& \pm \frac{\sqrt{25}}{\sqrt{169}}\hfill \\ \hfill \text{sin }t& =& \pm \frac{5}{13}\hfill \end{array}$$

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.

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

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As we discussed at the beginning of the chapter, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or $2\pi ,$ will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.

Other functions can also be periodic. For example, the lengths of months repeat every four years. If $x$ represents the length time, measured in years, and $f(x)$ represents the number of days in February, then $f(x+4)=f(x).$ This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.