Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent

Recall the following information that was covered in Section 3.1.

Consider a right triangle △ ABC, with the right angle at C and with lengths a, b, and c, as in the Figure 1 below. For the acute angle A, call the leg BC its opposite side, and call the leg AC its adjacent side. Recall that the hypotenuse of the triangle is always opposite the right angle.  In the triangle below, this is the side AB. The ratios of sides of a right triangle occur often enough in practical applications to warrant their own names, so we define the six trigonometric functions of A as follows:

Figure 1:  Sides of a right triangle with respect to angle A.

We will usually use the abbreviated names of the functions. Notice from Table 1 that the pairs sin(A) and csc(A), cos(A) and sec(A), and tan(A) and cot(A) are reciprocals:

Also recall the work we did in section 3.3 when we defined the sine and cosine functions in terms of the unit circle.  For any angle [latex]t,[/latex] we labeled the intersection of the terminal side and the unit circle as by its coordinates, [latex]\left(x,y\right).[/latex]   We considered an acute angle in the first quadrant and dropped a perpendicular line to the x- axis to create a right triangle. The sides of the right triangle were then [latex]x[/latex] and [latex]y.[/latex]   When we used our right trigonometric definitions above, we saw that [latex]\mathrm{cos}\left(t\right)=\frac{x}{1}[/latex] and [latex]\mathrm{sin}\left(t\right)=\frac{y}{1}.[/latex]  This means the ordered pair [latex]\left(x,y\right)=\left(\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right).[/latex]  See Figure 2 below.

As with the sine and cosine, we can use the [latex]\left(x,y\right)[/latex] coordinates to find the other functions.

Figure 2: Unit circle where the central angle is in radians

•  In right triangle trigonometry, the tangent of an angle is the ratio of the opposite side over the adjacent side with respect to the angle.  In Figure 2, the tangent of angle [latex]t[/latex] is equal to [latex]\frac{y}{x},\text{ where }x\ne0.[/latex]
• Because the y-value is equal to the sine of [latex]t,[/latex] and the x-value is equal to the cosine of [latex]t,[/latex] the tangent of angle [latex]t[/latex] can also be defined as [latex]\frac{\mathrm{sin}\left(t\right)}{\mathrm{cos}\left(t\right)},\text{ where }\mathrm{cos}\left(t\right)\ne0.[/latex]
• The remaining three functions can also all be expressed as functions of a point on the unit circle.
• When we change the y-value to the sine of [latex]t,[/latex] and the x-value to the cosine of [latex]t,[/latex] we can  express the functions in terms of the sine and cosine functions.  When we do this, we typically refer to these statements as basic trignometric identities.    See the definition box below for details.

Example 1:  Finding Trigonometric Functions from a Point on the Unit Circle

The point [latex]\left(-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2},\frac{1}{2}\right)[/latex] is on the unit circle, as shown in  Figure 3.   Find [latex]\mathrm{sin}\left(t\right),\text{ }\mathrm{cos}\left(t\right),\text{ }\mathrm{tan}\left(t\right),\text{ }\mathrm{sec}\left(t\right),\text{ }\mathrm{csc}\left(t\right),[/latex] and [latex]\mathrm{cot}\left(t\right).[/latex]

Figure 3:  Graph of circle with angle of t inscribed.

Show Solution

Because we know the [latex]\left(x,y\right)[/latex] coordinates of the point on the unit circle indicated by angle [latex]t,[/latex] we can use those coordinates to find the six functions.  First we know

[latex]\begin{align*}\mathrm{sin}\left(t\right)&=y=\frac{1}{2}\\ \mathrm{cos}\left(t\right)&=x=-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}\end{align*}[/latex]

Since [latex]\mathrm{tan}\left(t\right)=\frac{y}{x}[/latex] or [latex]\mathrm{tan}\left(t\right)=\frac{\mathrm{sin}\left(t\right)}{\mathrm{cos}\left(t\right)}[/latex] using the values for sine and cosine, we have

[latex]\mathrm{tan}\left(t\right)=\frac{y}{x}=\frac{\frac{1}{2}}{-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}}=\frac{1}{2}\left(-\frac{2}{\sqrt[\leftroot{1}\uproot{2} ]{3}}\right)=-\frac{1}{\sqrt[\leftroot{1}\uproot{2} ]{3}}=-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{3}.[/latex][latex]\\[/latex]

Since [latex]\mathrm{sec}\left(t\right)=\frac{1}{x}[/latex] or [latex]\mathrm{sec}\left(t\right)=\frac{1}{\mathrm{cos}\left(t\right)}[/latex] and we know the value for cosine, we have

[latex]\mathrm{sec}\left(t\right)=\frac{1}{-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}}=-\frac{2}{\sqrt[\leftroot{1}\uproot{2} ]{3}}=-\frac{2\sqrt[\leftroot{1}\uproot{2} ]{3}}{3}.[/latex][latex]\\[/latex]

Since  [latex]\mathrm{csc}\left(t\right)=\frac{1}{y}[/latex] or [latex]\mathrm{csc}\left(t\right)=\frac{1}{\mathrm{sin}\left(t\right)}[/latex] and we know the value for sine, we have

[latex]\mathrm{csc}\left(t\right)=\frac{1}{\frac{1}{2}}=2.[/latex][latex]\\[/latex]

Finally, since [latex]\mathrm{cot}\left(t\right)=\frac{x}{y}[/latex] or [latex]\mathrm{cot}\left(t\right)=\frac{\mathrm{cos}\left(t\right)}{\mathrm{sin}\left(t\right)}[/latex] using the values for sine and cosine, we have

[latex]\mathrm{cot}\left(t\right)=\frac{-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}}{\frac{1}{2}}=\frac{-\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}\left(\frac{2}{1}\right)=-\sqrt[\leftroot{1}\uproot{2} ]{3}.[/latex][latex]\\[/latex]

Try it #1

The point [latex]\left(\frac{\sqrt[\leftroot{1}\uproot{2} ]{2}}{2},-\frac{\sqrt[\leftroot{1}\uproot{2} ]{2}}{2}\right)[/latex] is on the unit circle, as shown in  Figure 4.

Find [latex]\mathrm{sin}\left(t\right),\text{ }\mathrm{cos}\left(t\right),\text{ }\mathrm{tan}\left(t\right),\text{ }\mathrm{sec}\left(t\right),\text{ }\mathrm{csc}\left(t\right),[/latex] and [latex]\mathrm{cot}\left(t\right).[/latex]

Figure 4:  Graph of circle with angle of t inscribed.

Show Solution

[latex]\mathrm{sin}\left(t\right)=-\frac{\sqrt[\leftroot{1}\uproot{2} ]{2}}{2},\text{ }\mathrm{cos}\left(t\right)=\frac{\sqrt[\leftroot{1}\uproot{2} ]{2}}{2},\text{ }\mathrm{tan}\left(t\right)=-1,\text{ }\mathrm{sec}\left(t\right)=\sqrt[\leftroot{1}\uproot{2} ]{2},\text{ }\mathrm{csc}\left(t\right)=-\sqrt[\leftroot{1}\uproot{2} ]{2},\text{ }\mathrm{cot}\left(t\right)=-1[/latex]

Try it #2

Find the values of the six trigonometric functions of angle [latex]t[/latex] based on Figure 5.

Figure 5:  Graph of circle with angle of t inscribed.

Show Solution

[latex]\begin{align*}\mathrm{sin}\left(t\right)&=-1,\\\mathrm{cos}\left(t\right)&=0,\\\mathrm{tan}\left(t\right)&=\text{Undefined}\\ \mathrm{sec}\left(t\right)&=\text{ Undefined,}\\\mathrm{csc}\left(t\right)&=-1,\\\mathrm{cot}\left(t\right)&=0\end{align*}[/latex]

Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting [latex]x[/latex] equal to the cosine and [latex]y[/latex] equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in Table 2.

Evaluating Trigonometric Functions with a Calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.

Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.

If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor [latex]\frac{\pi }{180}[/latex] to convert the degrees to radians. To find the secant of [latex]30°,[/latex] we could press

or

Analyzing the Graph of y = tan(x)

We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that

[latex]\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}.[/latex][latex]\\[/latex]

Remember that there are some values of [latex]x[/latex] for which [latex]\mathrm{cos}\left(x\right)=0.[/latex] For example, [latex]\mathrm{cos}\left(\frac{\pi }{2}\right)=0[/latex] and [latex]\mathrm{cos}\left(\frac{3\pi }{2}\right)=0.[/latex] At these values, the tangent function is undefined, so the graph of [latex]y=\mathrm{tan}\left(x\right)[/latex] has discontinuities at [latex]x=\frac{\pi }{2}[/latex] and [latex]\frac{3\pi }{2}.[/latex] We will examine the function from a numerical point of view to see if there is evidence that there are vertical asymptotes at these points of discontinuity.

We have already shown, using the ideas from the unit circle, that the tangent function is odd.

We can further analyze the numerical behavior of the tangent function by looking at values for some of the special angles, as listed in Table 4.

Table 4

[latex]x[/latex]	[latex]-\frac{\pi }{2}[/latex]	[latex]-\frac{\pi }{3}[/latex]	[latex]-\frac{\pi }{4}[/latex]	[latex]-\frac{\pi }{6}[/latex]	0	[latex]\frac{\pi }{6}[/latex]	[latex]\frac{\pi }{4}[/latex]	[latex]\frac{\pi }{3}[/latex]	[latex]\frac{\pi }{2}[/latex]
[latex]\mathrm{tan}\left(x\right)[/latex]	undefined	[latex]-\sqrt[\leftroot{1}\uproot{2} ]{3}[/latex]	–1	[latex]-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{3}[/latex]	0	[latex]\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{3}[/latex]	1	[latex]\sqrt[\leftroot{1}\uproot{2} ]{3}[/latex]	undefined

These points will help us draw our graph, but we need to determine how the graph behaves where the function is undefined. If we look more closely at values when [latex]\frac{\pi }{3}<x<\frac{\pi }{2},[/latex] we can use a table to look for a trend. Because [latex]\frac{\pi }{3}\approx 1.05[/latex] and [latex]\frac{\pi }{2}\approx 1.5707,[/latex] we will evaluate [latex]x[/latex] at radian measures [latex]1.05<x<1.5707[/latex] as shown in Table 5.

Table 5

[latex]x[/latex]	1.3	1.5	1.55	1.56	1.57
[latex]\mathrm{tan} \left(x\right)[/latex]	3.6	14.1	48.1	92.6	1255.8

 

As [latex]x[/latex] approaches [latex]\frac{\pi }{2}[/latex] from the left hand side, the outputs of the function get larger and larger or as [latex]x\to{\frac{\pi}{2}}^{-},\text{ }\mathrm{tan}\left(x\right)\to\infty.[/latex] This provides us with evidence that there is a vertical asymptote at [latex]\frac{\pi }{2}.[/latex]

Because [latex]y=\mathrm{tan}\left(x\right)[/latex] is an odd function, we see the corresponding table of negative values in Table 6.

Table 6

[latex]x[/latex]	−1.3	−1.5	−1.55	−1.56	-1.57
[latex]\mathrm{tan}\left(x\right)[/latex]	−3.6	−14.1	−48.1	−92.6	-1255.8

We can see that, as [latex]x[/latex] approaches [latex]-\frac{\pi }{2}[/latex] from the right hand side, the outputs get more and more negative [latex]x\to{\frac{-\pi}{2}}^{+},\text{ }\mathrm{tan}\left(x\right)\to-\infty.[/latex]  Again, this gives us evidence that these is a vertical asymptote at [latex]-\frac{\pi }{2}.[/latex]

Figure 9 represents the graph of [latex]y=\mathrm{tan}\left(x\right).[/latex] The tangent is positive from 0 to [latex]\frac{\pi }{2}[/latex] and from [latex]\pi [/latex] to [latex]\frac{3\pi }{2},[/latex] corresponding to quadrants I and III of the unit circle.

We could create more points for other intervals and we will see that the tangent function repeats its behavior every [latex]\pi[/latex] units.  We therefore conclude that the tangent function has a period of [latex]\pi[/latex].

Figure 9: Graph of the tangent function.

Analyzing the Graphs of y = sec(x) and y = csc(x)

The secant was defined by the reciprocal identity [latex]\mathrm{sec}\left(x\right)=\frac{1}{\mathrm{cos}\left(x\right)}.[/latex] Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\frac{\pi }{2},[/latex] [latex]\frac{3\pi }{2},[/latex] etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.

We can graph [latex]y=\mathrm{sec}\left(x\right)[/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure 10. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.

The secant graph has vertical asymptotes at each value of [latex]x[/latex] where the cosine graph crosses the x-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.

Note that, because cosine is an even function, secant is also an even function. That is, [latex]\mathrm{sec}\left(-x\right)=\mathrm{sec}\left(x\right)[/latex]

Figure 10: Graph of the secant function, [latex]f\left(x\right)=\mathrm{sec}\left(x\right)=\frac{1}{\mathrm{cos}\left(x\right).}[/latex]

Similar to the secant, the cosecant is defined by the reciprocal identity [latex]\mathrm{csc}\left(x\right)=\frac{1}{\mathrm{sin}}\left(x\right).[/latex] Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at [latex]0,[/latex] [latex]\pi,[/latex] etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.

We can graph [latex]y=\mathrm{csc}\left(x\right)[/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure 11. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.

The cosecant graph has vertical asymptotes at each value of [latex]x[/latex] where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines.

Note that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\mathrm{csc}\left(-x\right)=\mathrm{-csc}\left(x\right).[/latex]

The graph of cosecant, which is shown in Figure 9, is similar to the graph of secant.

Figure 11: The graph of the cosecant function, [latex]f\left(x\right)=\mathrm{csc}\left(x\right)=\frac{1}{\mathrm{sin}\left(x\right).}[/latex]

Analyzing the Graph of y = cot(x)

The last trigonometric function we need to explore is cotangent. The cotangent is defined by the reciprocal identity [latex]\mathrm{cot}\left(x\right)=\frac{1}{\mathrm{tan}\left(x\right)}.[/latex] Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at [latex]0,\pi ,[/latex] etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.

We can graph [latex]y=\mathrm{cot}\left(x\right)[/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. See Figure 12. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.

The cotangent graph has vertical asymptotes at each value of [latex]x[/latex] where [latex]\mathrm{tan}\left(x\right)=0;[/latex] we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\mathrm{cot}\left(x\right)[/latex] has vertical asymptotes at all values of [latex]x[/latex] where [latex]\mathrm{tan}\left(x\right)=0,[/latex] and [latex]\mathrm{cot}\left(x\right)=0[/latex] at all values of [latex]x[/latex] where [latex]\mathrm{tan}\left(x\right)[/latex] has its vertical asymptotes.

Figure 12: The cotangent function.

Period of a Function

As we have previously discussed, a function that repeats its values in regular intervals is known as a periodic function.  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.

Remember, 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] 

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.

Access these online resources for additional instruction and practice with other trigonometric functions.

• Determining Trig Function Values
• More Examples of Determining Trig Functions
• Pythagorean Identities
• Trig Functions on a Calculator