SOLUTION: A function f is defined by f(x)= cos(^2)x + 1/csc2x. What is f(pi/6)?

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Question 612804: A function f is defined by f(x)= cos(^2)x + 1/csc2x. What is f(pi/6)?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
function is defined as:

f(x) = cos^2(x) + 1/csc(2x)

what is f(pi/6)

you replace x with pi/6 and solve.

cos^2(pi/6) = [cos(pi/y)]^2= (.866025)^2 = .75
1/csc(2x) = sin(2x) = sin(2pi/6) = .8660254038

cos^2(pi/6) + 1/csc(2x) = .75 + .86... = 1.616025404

if you convert this to degrees it might be easier to see.

you convert pi/6 to degrees by multiplying it by 180 and dividing it by pi.

pi/6 * 180/pi = 180/6 = 30 degrees.

cosine 30 degrees = sqrt(3)/2

sine 60 degrees = sqrt(3)/2

cos^2(30) = (sqrt(3)/2)^2 = 3/4

1/csc(2x) = sin(2x) because sine = 1 / cosecant and cosecant = 1 / sine.

x = 30 so 2x = 60 degrees.

sine(60) = sqrt(3)/2

cos^2(30) + sin(60) = 3/4 + sqrt(3)/2 = (3 + 2*sqrt(3))/4 = 1.616025404.

this is the same answer we got before using radians.

the graph of the equation of cos^2(x) + 1/csc(2x) is shown below.

getting it to graph properly is a challenge, but if you know a few tricks, it can be done.

first trick:

cos^2(x) is the same as (cos(x))^2
first you find cos(x) and then you square it.

second trick:

csc(2x) = 1/sin(2x) and sin(2x) = 1/csc(2x) by definition.
sin(2x) is the sine of 2 times the angle.
if the angle is 30 degrees, then 2x is 60 degrees.

last, but not least, is that the graphing software deals in radians and not degrees.

on the graphing software, the value of x = pi/6 radians will show up as .523598776 because pi/6 is equal to 3.141592654... / 6 which is equal to .5235.....

if you're using a scientific calculator, make sure to set it to radians before trying to solve this problem directly.

if you want to work with degrees, make sure to convert the radians to degrees before working the problem.