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Use reference angles to find all six trigonometric functions of 7 π 4 .

sin ( 7 π 4 ) = 2 2 , cos ( 7 π 4 ) = 2 2 , tan ( 7 π 4 ) = 1 , sec ( 7 π 4 ) = 2 , csc ( 7 π 4 ) = 2 , cot ( 7 π 4 ) = 1

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Using even and odd trigonometric functions

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

Consider the function f ( x ) = x 2 , shown in [link] . The graph of the function is symmetrical about the y -axis. All along the curve, any two points with opposite x -values have the same function value. This matches the result of calculation: ( 4 ) 2 = ( −4 ) 2 , ( −5 ) 2 = ( 5 ) 2 , and so on. So f ( x ) = x 2 is an even function, a function such that two inputs that are opposites have the same output. That means f ( x ) = f ( x ) .

This is an image of a graph of and upward facing parabola with points (-2, 4) and (2, 4) labeled.
The function f ( x ) = x 2 is an even function.

Now consider the function f ( x ) = x 3 , shown in [link] . The graph is not symmetrical about the y -axis. All along the graph, any two points with opposite x -values also have opposite y -values. So f ( x ) = x 3 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f ( x ) = f ( x ) .

This is an image of a graph of the function f of x = x to the third power with labels for points (-1, -1) and (1, 1).
The function f ( x ) = x 3 is an odd function.

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in [link] . The sine of the positive angle is y . The sine of the negative angle is y . The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in [link] .

Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.
sin  t = y sin ( t ) = y sin  t sin ( t ) cos  t = x cos ( t ) = x cos  t = cos ( t ) tan ( t ) = y x tan ( t ) = y x tan  t tan ( t )
sec  t = 1 x sec ( t ) = 1 x sec  t = sec ( t ) csc  t = 1 y csc ( t ) = 1 y csc  t csc ( t ) cot  t = x y cot ( t ) = x y cot  t cot ( t )

Even and odd trigonometric functions

An even function is one in which f ( x ) = f ( x ) .

An odd function is one in which f ( x ) = f ( x ) .

Cosine and secant are even:

cos ( t ) = cos  t sec ( t ) = sec  t

Sine, tangent, cosecant, and cotangent are odd:

sin ( t ) = sin  t tan ( t ) = tan  t csc ( t ) = csc  t cot ( t ) = cot  t

Using even and odd properties of trigonometric functions

If the secant of angle t is 2, what is the secant of t ?

Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is 2, the secant of t is also 2.

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If the cotangent of angle t is 3 , what is the cotangent of t ?

3

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Recognizing and using fundamental identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity    we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

Fundamental identities

We can derive some useful identities    from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

tan t = sin t cos t
sec t = 1 cos t
csc t = 1 sin t
cot t = 1 tan t = cos t sin t

Questions & Answers

1KI POWER 1/3 PLEASE SOLUTIONS
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Amit
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Dorbor
well
Biswajit
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Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
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siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
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Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
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Neha
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Mirza
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Khamis
exact value of cos(π/3-π/4)
Ankit Reply
Practice Key Terms 6

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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