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Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.
Evaluate $\text{\hspace{0.17em}}\text{csc}\left(\frac{7\pi}{6}\right).$
$-2$
Simplify $\text{\hspace{0.17em}}\frac{\mathrm{sec}\text{\hspace{0.17em}}t}{\mathrm{tan}\text{\hspace{0.17em}}t}.$
We can simplify this by rewriting both functions in terms of sine and cosine.
By showing that $\text{\hspace{0.17em}}\frac{\mathrm{sec}\text{\hspace{0.17em}}t}{\mathrm{tan}\text{\hspace{0.17em}}t}\text{\hspace{0.17em}}$ can be simplified to $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}t,$ we have, in fact, established a new identity.
Simplify $\text{\hspace{0.17em}}(\mathrm{tan}\text{\hspace{0.17em}}t)(\mathrm{cos}\text{\hspace{0.17em}}t).$
$\mathrm{sin}t$
We can use these fundamental identities to derive alternate forms of the Pythagorean Identity, $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1.\text{\hspace{0.17em}}$ One form is obtained by dividing both sides by $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t.$
The other form is obtained by dividing both sides by $\text{\hspace{0.17em}}{\mathrm{sin}}^{2}t.$
If $\text{\hspace{0.17em}}\mathrm{cos}(t)=\frac{12}{13}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1,$ and the remaining functions by relating them to sine and cosine.
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, $\text{\hspace{0.17em}}-\frac{5}{13}.$
The remaining functions can be calculated using identities relating them to sine and cosine.
If $\text{\hspace{0.17em}}\mathrm{sec}(t)=-\frac{17}{8}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0<t<\pi ,$ find the values of the other five functions.
$\begin{array}{l}\mathrm{cos}t=-\frac{8}{17},\text{}\mathrm{sin}t=\frac{15}{17},\text{}\mathrm{tan}t=-\frac{15}{8}\\ \mathrm{csc}t=\frac{17}{15},\text{}\mathrm{cot}t=-\frac{8}{15}\end{array}$
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 $\text{\hspace{0.17em}}2\pi ,$ 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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the length time, measured in years, and $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ represents the number of days in February, then $\text{\hspace{0.17em}}f(x+4)=f(x).$ 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.
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