find the exact value?

Question

sin 450°, cos(-780°), and tan (37π/6)

Answer 1

1. sin 450° = sin 90° if we subtract one revolution, or 360°, to find the co-terminal angle. We know that sin 90° = 1

 

2. cos (-780°) = cos (-60°) when we add two revolutions, or 720°. In general:

 

cos (-x) = cos x, so cos (-60°) = cos 60° = 1/2

 

3. tan (37π/6) = tan (π/6) when we subtract three revolutions, or 36π/6 radians. Remember 2π radians = one revolution.

 

tan (π/6) = 1/√3 = √3/3

 

Answer 2

Rewrite the angle as a sum of angles that we can easily find sine, cosine, and tangent of.

 

sin(450) = sin(360 + 90)

 

You know that 360 degrees is on the x-axis on the unit circle.  If we rotate 90 degrees counterclockwise from this position, we are at the positive y-axis.  The y-coordinate at the positive y-axis is 1.

 

sin(450) = 1

 

You can also apply the addition angle identity for sine.

 

 

cos(-780) = cos(-720 - 60)

 

-720 degrees is on the positive x-axis (because 360 + 360 = 720).  This rotation is 2 complete circles overlapping each other.  If we rotate 60 degrees clockwise from this position, we will be in the 4th quadrant.  We know the cosine is positive in the 4th quadrant and cos(60) is 1/2.

 

cos(-780) = 1/2

 

 

 

tan(37π/6) = tan(36π/6 + π/6)

                 = tan(6π + π/6)

 

 

6π is on the positive x-axis.  If we rotate π/3 counterclockwise from this position, we will be in the 1st quadrant.

 

tan(π/6) = sin(π/6) / cos(π/6) = (1/2) * (2/√3) = 1/√3 = √(3)/3

 

tan(37π/6) = √(3)/3 

 

 

Note:  We could have use the addition and subtraction angle identities for all of these.