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.