We can use these fundamental identities to derive alternative forms of the Pythagorean Identity, [latex]{\cos }^{2}t+{\sin }^{2}t=1[/latex]. One form is obtained by dividing both sides by [latex]{\cos }^{2}t:[/latex]
If [latex]\text{cos}\left(t\right)=\frac{12}{13}[/latex] and [latex]t[/latex] is in quadrant IV, as shown in Figure 8, find the values of the other five trigonometric functions. Figure 8
We can find the sine using the Pythagorean Identity, [latex]{\cos }^{2}t+{\sin }^{2}t=1[/latex], 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, [latex]-\frac{5}{13}[/latex].
The remaining functions can be calculated using identities relating them to sine and cosine.
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 [latex]2\pi[/latex], 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 [latex]x[/latex] represents the length time, measured in years, and [latex]f\left(x\right)[/latex] represents the number of days in February, then [latex]f\left(x+4\right)=f\left(x\right)[/latex]. 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.
period of a function
The period [latex]P[/latex] of a repeating function [latex]f[/latex] is the number representing the interval such that [latex]f\left(x+P\right)=f\left(x\right)[/latex] for any value of [latex]x[/latex].
The period of the cosine, sine, secant, and cosecant functions is [latex]2\pi[/latex].
The period of the tangent and cotangent functions is [latex]\pi[/latex].
Find the values of the six trigonometric functions of angle [latex]t[/latex] based on Figure 9. Figure 9
Find the values of the six trigonometric functions of angle [latex]t[/latex] based on Figure 10.
Figure 10
[latex]\begin{align}&\sin t=-1\\&\cos t=0\\&\tan t \text{ is undefined}\\ &\sec t \text{ is undefined}\\&\csc t=-1\\&\cot t=0\end{align}[/latex]
If [latex]\sin \left(t\right)=-\frac{\sqrt{3}}{2}[/latex] and [latex]\text{cos}\left(t\right)=\frac{1}{2}[/latex], find [latex]\text{sec}\left(t\right),\text{csc}\left(t\right),\text{tan}\left(t\right),\text{ cot}\left(t\right)[/latex].
If [latex]\sin \left(t\right)=\frac{\sqrt{2}}{2}[/latex] and [latex]\cos \left(t\right)=\frac{\sqrt{2}}{2}[/latex], find [latex]\text{sec}\left(t\right),\text{csc}\left(t\right),\text{tan}\left(t\right),\text{ and cot}\left(t\right)[/latex].
