Use an Addition or Subtraction Formula to find the exact value of the expression: tan 285°

Question

I am so confused

Answer 1

285° is not an angle that has a simple value for any of the trigonometric functions.  But if you can find two angles that add to 285 that DO have simple values, then you can use the addition formula for tangent to find the answer.

 

What are the angles that have simple values?  You've probably seen this already.  They are:

 

In the first quadrant, they are 0, 30, 45, 60, and 90.  And by the rules for finding the function values as you go around the unit circle, you can add 90, 180 and 270 to these to find angles in the other quadrants that have simple values.  So the full set of angles that have simple values is:

 

Quadrant I:  0, 30, 45, 60, and 90

Quadrant II: 120, 135, 150, 180

Quadrant III:  210, 225, 240, 270

Quadrant IV:  300, 315, 330, 360

 

So how can you take two of these to add up to 285?  There are several ways.  Let's use 45 and 240.

 

Now look at the sum formula for tangent:

 

tan (a + b) = (?1 + tan a tan b) / (??tan a - tan b)

 

So you need to find tan 45 and tan 240.  tan 45 is easy, that's just 1.  Since the period of tan is 180, tan 240 is the same as tan 60, ??which is √3.  So plug these into the formula:

 

tan (45 + 240) = (?1 + tan 45 tan 240) / (??tan 45 - tan 240) =

(1 + 1*√3) / (1 - √3) =

(1 + √3) / (1 - √3)

 

You can "rationalize the denominator" by multiplying by (1 + √3) / (1 + √3).  That gives you:

 

[(1 + √3)(1 + √3)] / [(1 - √3) (1 + √3)] =

(1 + 2√3 + 3) / (1 - 3) =

(4 + 2√3) / (-2) =

-(2 + √3) =

-2 - √3