<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

3

Got questions? Get instant answers now!

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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask