<PRE><B>Question 1186660</B>
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&#39;s what it looks like on the unit circle graph.


&lt;img src = &quot;http://theo.x10hosting.com/2021/102501.jpg&quot; &gt;


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&#39;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&#39;s because 270 + 30 = 300 and you get:


cos(300) = sin(30).



















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