Use an Addition or Subtraction Formula to find the exact value of the expression: tan 105°

Question

I need help

Answer 1

tan(105) = sin(105) / cos(105)

 

 

Find two angles that give a sum or difference of 105.  These angles should be values that you can easily identify sine and cosine of.  Then use the addition or subtraction formula.

 

tan(60 + 45) = (sin(60 + 45)) / (cos(60 + 45))

 

Lets evaluate sin(60 + 45) first.  Then evaluate cos(60 + 45).  After that, divide the results.

 

 

sin(60 + 45) = sin(60)cos(45) + cos(60)sin(45)

                   = (√(3) / 2)(√(2) / 2) + (1 / 2)(√(2) / 2)

                   = (√(6) + √(2)) / 4

 

cos(60 + 45) = cos(60)cos(45) - sin(60)sin(45)

                    = (1 / 2)(√(2) / 2) - (√(3) / 2)(√(2) / 2)

                    = (√(2) - √(6)) / 4

 

Now, divide the sine value by the cosine value.  You will end up with

 

(√(6) + √(2)) / (√(2) - √(6))

 

 

Multiply the expression by (√(2) + √(6)) / (√(2) + √(6)) to rationalize the denominator.  Simplify and end with

 

(8 + 4√3) / (-4)

 

 

Simplify the expression to obtain

 

-2 - √3