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Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.

  1. If the graphing utility has degree mode and radian mode, set it to radian mode.
  2. Enter: 1  /
  3. Press the SIN key.
  4. Enter the value of the angle inside parentheses.
  5. Press the ENTER key.

Evaluating the secant using technology

Evaluate the cosecant of 5 π 7 .

For a scientific calculator, enter information as follows:

1 / ( 5  ×   π  / 7 ) SIN =
csc ( 5 π 7 ) 1.279
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Evaluate the cotangent of π 8 .

2.414

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

Key equations

Tangent function tan t = sin t cos t
Secant function sec t = 1 cos t
Cosecant function csc t = 1 sin t
Cotangent function cot t = 1 tan t = cos t sin t

Key concepts

  • The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle.
  • The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
  • The six trigonometric functions can be found from a point on the unit circle. See [link] .
  • Trigonometric functions can also be found from an angle. See [link] .
  • Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See [link] .
  • A function is said to be even if f ( x ) = f ( x ) and odd if f ( x ) = f ( x ) .
  • Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
  • Even and odd properties can be used to evaluate trigonometric functions. See [link] .
  • The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
  • Identities can be used to evaluate trigonometric functions. See [link] and [link] .
  • Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See [link] .
  • The trigonometric functions repeat at regular intervals.
  • The period P of a repeating function f is the smallest interval such that f ( x + P ) = f ( x ) for any value of x .
  • The values of trigonometric functions of special angles can be found by mathematical analysis.
  • To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See [link] .

Section exercises

Verbal

On an interval of [ 0 , 2 π ) , can the sine and cosine values of a radian measure ever be equal? If so, where?

Yes, when the reference angle is π 4 and the terminal side of the angle is in quadrants I and III. Thus, at x = π 4 , 5 π 4 , the sine and cosine values are equal.

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What would you estimate the cosine of π degrees to be? Explain your reasoning.

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For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Substitute the sine of the angle in for y in the Pythagorean Theorem x 2 + y 2 = 1. Solve for x and take the negative solution.

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Describe the secant function.

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Tangent and cotangent have a period of π . What does this tell us about the output of these functions?

The outputs of tangent and cotangent will repeat every π units.

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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
lurverkitten
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.
Liam
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
Ama
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.
Spiro
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
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
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
Abena
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
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
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.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"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?
Spiro
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
abdikarin
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
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
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
William
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.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
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."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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