SOLUTION: For the following trigonometric function, identify whether the identity stated is true or false. If it is true, demonstrate this with a diagram on the unit circle. if it is false,

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: For the following trigonometric function, identify whether the identity stated is true or false. If it is true, demonstrate this with a diagram on the unit circle. if it is false,       Log On


   

Question 1186660: For the following trigonometric function, identify whether the identity stated is true or false. If it is true, demonstrate this with a diagram on the unit circle. if it is false, identify what the expression on the left actually is :
Cos(3pi/2 + x) = sin(x)
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the identity you are looking for is:

cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b)

your problem states that:

cos(3pi/2 + x) = sin(x)

if you let a = 3pi/2 and b = x, then your identity equation of:

cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b) becomes:

cos(3pi/2 + x) = cos(3pi/2) * cos(x) - sin(3pi/2) * sin(x).

you can use your calculator to find::

cos(3pi/2) = 0

sin(3pi/2) = -1

make sure you set your calculator to radians when you do this.

if you are not comfortable working with radians, then convert 3pi/2 radians to degrees by multiplying it by 180 / pi.

3pi/2 * 180/pi = 3/2 * 180 = 270 degrees.

set your calculator to degrees and you will find that:

cos(270) = 0

sin(270) = -1

these are the same value when the angle is in degrees and when the angle is in radians.

the conversion factor is:

degrees = radians * 180 / pi.

radians = degrees * pi / 180.

in your identity equation of:

cos(3pi/2 + x) = cos(3pi/2) * cos(x) - sin(3pi/2) * sin(x).

replace cos(3pi/2) with 0 and replace sin(3pi/2) with -1 to get:

cos(3pi/2 + x) = 0 * cos(x) - (-1 * sin(x).

simplify to get:

cos(3pi/2 + x) = 0 + 1 * sin(x).

simplify further to get:

cos(3pi/2 + x) = sin(x)

this confirms the identity is good.

you can demonstrate this is true on the unit circle by showing an example:

x = 30 degrees.

270 + x = 300 degrees.

the reference angle for 300 degrees is equal to 360 - 300 = 60 degrees.

here's what it looks like on the unit circle graph.



the 300 degree angle is equal to 270 + 30.

the 300 degree angle is also equal to 360 - 60.

the 300 degree angle forms the triangle ABD.

the reference angle for the 300 degree angle is equal to 60 degrees.

that's angle DAB in the diagram.

cos(300) is equal to adjadcent side divided by hypotenuse.

hypotenuse is equal to 1.
adjacent side is AD = .5
cos(300) is therefore equal to .5/1 = .5.

sin(30) is equal to .5 as well.

therefore, if x = 30 degrees, then cos(270 + 30) = sin(30).

that's because 270 + 30 = 300 and you get:

cos(300) = sin(30).