find exact values of the trigonometric features secant, cosecant, tangent, and also cotangent of (fracpi3), (fracpi4), and also (fracpi6). Use referral angles to evaluate the trigonometric attributes secant, tangent, and cotangent. Use properties that even and also odd trigonometric functions. Recognize and also use an essential identities. Advice trigonometric attributes with a calculator.

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A wheelchair ramp the meets the standards of the Americans with Disabilities Act should make an angle with the ground whose tangent is (frac112) or less, regardless of that length. A tangent represents a ratio, for this reason this method that because that every 1 inch of rise, the ramp must have actually 12 customs of run. Trigonometric functions permit us come specify the shapes and proportions that objects independent of exact dimensions. Us have currently defined the sine and also cosine functions of an angle. Despite sine and cosine room the trigonometric functions most often used, over there are four others. Together they make up the collection of 6 trigonometric functions. In this section, we will certainly investigate the continuing to be functions.

Finding precise Values of the Trigonometric attributes Secant, Cosecant, Tangent, and Cotangent

To specify the continuing to be functions, us will once again draw a unit circle v a point ((x,y)) corresponding to an angle of (t),as displayed in figure (PageIndex1). Similar to the sine and cosine, we can use the ((x,y)) coordinates to find the other functions.


Exercise (PageIndex1):

The allude ((fracsqrt22,−fracsqrt22)) is top top the unit circle, as displayed in number (PageIndex3). Uncover ( sin t, cos t, an t, sec t, csc t,) and also ( cot t).

Figure (PageIndex3)


( sin t=−fracsqrt22, cos t= fracsqrt22, an t=−1, sec t=sqrt2, csc t=−sqrt2, cot t=−1)

Exercise (PageIndex2):

Find ( sin t, cos t, an t, sec t, csc t,) and ( cot t) as soon as (t=fracπ3.)


(eginalign sin fracπ3 & = fracsqrt32 \ cos fracπ3 &=frac12 \ an fracπ3 &= sqrt3 \ sec fracπ3 &= 2 \ csc fracπ3 &= frac2sqrt33 \ cot fracπ3 &= fracsqrt33 endalign)

Using recommendation Angles to evaluate Tangent, Secant, Cosecant, and Cotangent

We deserve to evaluate trigonometric features of angles exterior the an initial quadrant using recommendation angles together we have currently done through the sine and also cosine functions. The procedure is the same: find the reference angle created by the terminal next of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the exact same as those for the recommendation angle, other than for the positive or an unfavorable sign, i beg your pardon is figured out by x- and y-values in the initial quadrant. Number (PageIndex4) reflects which attributes are confident in i m sorry quadrant.

To help us remember i m sorry of the 6 trigonometric features are positive in every quadrant, we can use the mnemonic expression “A clever Trig Class.” every of the four words in the phrase coincides to among the 4 quadrants, beginning with quadrant I and also rotating counterclockwise. In quadrant I, which is “A,” all that the six trigonometric functions are positive. In quadrant II, “Smart,” just sine and also its mutual function, cosecant, space positive. In quadrant III, “Trig,” only tangent and also its mutual function, cotangent, room positive. Finally, in quadrant IV, “Class,” only cosine and its mutual function, secant, room positive.

Figure (PageIndex4)

Example (PageIndex3): Using referral Angles to discover Trigonometric Functions

Use recommendation angles to discover all 6 trigonometric functions of (−frac5π6).


The angle in between this angle’s terminal side and also the x-axis is (fracπ6), so that is the reference angle. Due to the fact that (−frac5π6) is in the 3rd quadrant, wherein both (x) and (y) are negative, cosine, sine, secant, and cosecant will be negative, when tangent and also cotangent will certainly be positive.

< eginalign cos (−dfrac5π6) &=−dfracsqrt32, sin (−dfrac5π6)=−dfrac12, an (−dfrac5π6) = dfracsqrt33 \ sec (−dfrac5π6) &=−dfrac2sqrt33, csc (−dfrac5π6)=−2, cot (−dfrac5π6)=sqrt3 endalign >

Using Even and also Odd Trigonometric Functions

To have the ability to use our six trigonometric functions freely v both positive and an unfavorable angle inputs, we should examine how each function treats a an adverse input. As it turns out, there is an important difference among the functions in this regard.

Consider the role (f(x)=x^2), displayed in figure (PageIndex5). The graph the the role is symmetrical about the y-axis. All along the curve, any type of two points v opposite x-values have actually the same role value. This matches the result of calculation: ((4)^2=(−4)^2,(−5)^2=(5)^2), and so on. For this reason (f(x)=x^2) is one even function, a function such that 2 inputs that are opposites have the same output. That method (f(−x)=f(x)).

Figure (PageIndex5): The function (f(x)=x^2) is an even function.

Now think about the function (f(x)=x^3), shown in figure (PageIndex6). The graph is not symmetrical around the y-axis. All follow me the graph, any type of two points through opposite x-values additionally have opposite y-values. Therefore (f(x)=x^3) is an odd function, one together that two inputs that room opposites have outputs the are likewise opposites. That way (f(−x)=−f(x)).

Figure (PageIndex6): The duty (f(x)=x^3) is an odd function.

We have the right to test even if it is a trigonometric role is even or odd by illustration a unit circle v a positive and also a an adverse angle, as in figure (PageIndex7). The sine of the optimistic angle is (y). The sine that the negative angle is −y. The sine function, then, is an odd function. We can test every of the six trigonometric functions in this fashion. The outcomes are displayed in Table (PageIndex2).

Figure (PageIndex7) Table (PageIndex2)
(eginalign sin t &=y \ sin (−t) &=−y \ sin t &≠sin(−t) endalign) ( eginalign cos t &=x \ cos (−t)=x \ cos t &= cos (−t) endalign) (eginalign an (t) &= fracyx \ an (−t) &=−fracyx \ an t &≠ an (−t) endalign)
(eginalign sec t &= frac1x \ sec (−t) &= frac1x \ sec t &= sec (−t) endalign) ( eginalign csc t &= frac1y \ csc (−t) &= frac1−y \ csc t &≠ csc (−t) endalign) ( eginalign cot t &= fracxy \ cot (−t) &= fracx−y \ cot t & ≠ cot (−t) endalign)


one even function is one in i beg your pardon (f(−x)=f(x)). One odd function is one in i m sorry (f(−x)=−f(x)).

Cosine and also secant room even:

< eginalign cos (−t) &= cos t \ sec (−t) &= sec t endalign>

Sine, tangent, cosecant, and also cotangent are odd:

<eginalign sin (−t) &=− sin t \ an (−t) &=− an t \ csc (−t) &=−csc t \ cot (−t) &=−cot t endalign>

Example (PageIndex4): making use of Even and also Odd properties of Trigonometric Functions

If the secant of angle t is 2, what is the secant the (−t)?


Secant is an also function. The secant that an edge is the exact same as the secant of its opposite. Therefore if the secant of angle t is 2, the secant the (−t) is additionally 2.

Example (PageIndex5): utilizing Identities to advice Trigonometric Functions

offered ( sin (45°)= fracsqrt22, cos (45°)= fracsqrt22), evaluate ( an(45°).) offered ( sin (frac5π6)= frac12, cos( frac5π6)=−fracsqrt32,) evaluate (sec (frac5π6)).


Because we understand the sine and also cosine worths for these angles, we have the right to use identities to advice the various other functions.

< eginalign* an(45°) &=dfrac sin(45°) cos (45°) \ &= dfracfracsqrt22fracsqrt22 \ & =1 endalign* > <eginalign* sec (dfrac5π6) &= dfrac1 cos (frac5π6) \ &= dfrac1−fracsqrt32 \ &= dfrac−2sqrt31 \ &=dfrac−2sqrt3 \ &=−dfrac2sqrt33 endalign*>

Exercise (PageIndex6)

Simplify (( an t)( cos t).)


( sin t )


<1+ an ^2 t= sec ^2 t >

< cot ^2 t+1= csc ^2 t>

Example (PageIndex7): utilizing Identities to Relate Trigonometric Functions

If cos(t)=1213 cos(t)=1213 and also t t is in quadrant IV, as presented in number (PageIndex8), discover the values of the other 5 trigonometric functions.


Exercise (PageIndex8)

Find the values of the six trigonometric functions of edge (t) based on Figure (PageIndex10).

Figure (PageIndex10)


(eginalign sin t &=−1, cos t=0, an t= extUndefined \ \sec t &= extUndefined, csc t=−1, cot t=0 endalign)

Example (PageIndex9): finding the worth of Trigonometric Functions

If ( sin(t)=−fracsqrt32) and also ( cos (t)=frac12),find ( sec (t),csc (t), an (t), cot (t).)


< eginalign sec t &= dfrac1 cos t= dfrac1frac12=2 \ csc t &= dfrac1 sin t= dfrac1−fracsqrt32−dfrac2sqrt33 \ an t &= dfracsin tcos t=dfrac−fracsqrt32frac12=−sqrt3 \ cot t &= dfrac1 an t= dfrac1−sqrt3=−dfracsqrt33 endalign>

Exercise (PageIndex9):

If (sin (t)= fracsqrt22) and also (cos (t)=fracsqrt22,) find ( sec (t), csc (t), an (t),) and also ( cot (t)).


( sec t= sqrt2,csc t=sqrt2, an t=1, cot t=1)

Evaluating Trigonometric functions with a Calculator

We have actually learned just how to advice the 6 trigonometric attributes for the usual first-quadrant angles and also to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of various other angles, we use a scientific or graphing calculator or computer system software. If the calculator has a level mode and a radian mode, check the correct setting is chosen before making a calculation.

Evaluating a tangent function with a clinical calculator as opposed to a graphing calculator or computer system algebra device is like evaluating a sine or cosine: go into the value and also press the TAN key. Because that the mutual functions, there might not be any specialized keys the say CSC, SEC, or COT. In that case, the duty must it is in evaluated together the reciprocal of a sine, cosine, or tangent.

If we should work through degrees and also our calculator or software does not have actually a degree mode, us can enter the levels multiplied through the conversion variable (fracπ180) to convert the degrees to radians. To find the secant the ( 30°), we can press


< aramuseum.orgrm(for ; a ; graphing ; calculator): dfrac1cos(frac30π180) >

how to: provided an angle measure up in radians, usage a graphing utility/calculator to find the cosecant

If the graphing energy has degree mode and also radian mode, collection it come radian mode. Enter: (1 ; /) push the SIN key. Go into the worth of the angle inside parentheses. Push the get in key.

Example (PageIndex10): examining the Cosecant making use of Technology

Evaluate the cosecant of (frac5π7).

See more: Device That Converts Ac To Dc Is Called ______? The Device Which Converts Ac To Dc Is


For a scientific calculator, go into information as follows:

< aramuseum.orgrm1 / ( 5 × π / 7 ) SIN =>

< aramuseum.orgrm csc (dfrac5π7)≈1.279 >


Access this online resources for extr instruction and practice with other trigonometric functions.