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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 ) .

Graph of 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 ) .

Graph of function 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 c o t ( 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 ?


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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.

Questions & Answers

can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
All real x except 5 and - 3
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
By using some imaginary no.
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
ok, one moment
how do I post your graph for you?
it won't let me send an image?
also for the first one... y=mx+b so.... y=3x-2
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Please were did you get y=mx+b from
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
thanks Tommy
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
consider r(a+b) = ra + rb. The a and b are the trig identity.
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
Practice Key Terms 6

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