Find the exact values of the cosine and sine of each angle. Then find the decimal equivalents. -225 degrees

Question

Can someone please explain how to find the exact values of the cosine and sine of 225 degrees? What is the first step?

Answer 1

We can also use the trig identities.

 

 

sin(-225)    can be written as a addition and subtraction angle identity.   Think of 2 angles we can easily find sine and cosine of on a unit circle, and the sum of the angle is -225.

 

270 and 45  work because    45 - 270 = -225

 

 

Therefore, we have a subtraction angle identity.

 

sin(-225) = sin(45 - 270)

 

               = sin(45)cos(270) - cos(45)sin(270)

 

               = 0 - (-1 * √(2) / 2)

 

                = √(2) / 2

 

 

sin(-225) = √(2) / 2)

 

 

Next, use the identity      sin²(x) + cos²(x) = 1       to find cos(-225).

 

 

cos(-225) = √(1 - sin²(-225))

 

               = √(1 - 2/4))

 

               = √(1 - 1/2)

 

               = √(1/2)

 

               = √(2) / 2

 

 

However, cosine is negative in the third quadrant, so

 

cos(-225) = -√(2) / 2

 

 

 

Answer 2

 

-225 degrees produces a coterminal angle of (360 -225). or 135 degrees.

 

180 - 135 = 45 degrees, so this produces a 45 degree reference angle in quadrant 2.

 

The side measures of the legs are both √2/2 ... x being negative in this quadrant.

 

Sine is the y-value ... which is √2/2

 

Cosine is the x-value ... which is -√2/2