# Find exact value of Trig function using Angle Addition or Subtraction

#### programmerBlack

##### New member
Hello,

I'm working on a problem (not-homework), that requires me to find the exact value of tan(15 degrees) using angle addition or subtraction identities. I have found the answer to be sqrt(3) - 3 / sqrt(3) + 3. I found this answer through trial and error. The first one I tried was was using tan (45 - 30), which yielded -1 (wrong) on Khan Academy. The second one I tried was tan(60 - 45), correct on Khan Academy.

My main question is this: How do I properly choose the angles so that my answer is correct the first time? Attached is a sample of my work. Thank you in advance.

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#### MarkFL

##### Super Moderator
Staff member
Let's try the first difference you tried:

$$\displaystyle \tan\left(15^{\circ}\right)=\tan\left(45^{\circ}-30^{\circ}\right)=\frac{\tan\left(45^{\circ}\right)-\tan\left(30^{\circ}\right)}{1+\tan\left(45^{\circ}\right)\tan\left(30^{\circ}\right)}=\frac{1-\dfrac{1}{\sqrt{3}}}{1+\dfrac{1}{\sqrt{3}}}=\frac{\sqrt{3}-1}{\sqrt{3}+1}$$

This is equivalent to the value you cite as the correct answer.

#### Dr.Peterson

##### Elite Member
Have you found the sign error at the end of the first long line in your work?

It wasn't that you chose the wrong two angles; you just chose angles that made the work complicated enough to hide a mistake.

#### programmerBlack

##### New member
Thank you for the fast replies. I can see that my error came from me doing my calculations incorrect. Which led me to believe that I needed to "choose some special angles" which is an incorrect assumption.

Thank you again.