tan(165 degrees)

Question

Find the exact values ​​using the addition or subtraction formulas.

Please show me a process, I dont understand how to do it.

Answer 1

Hi Joseph R.

For 165 degrees you have a choice of values that add or subtract to give 165 degrees they include

90 + 75

or

180 - 15

For a 165 degree angle you want to use the subtraction formula since the tan of 90 degrees is undefined, remember since the tangent is the sine divided by the cosine (sin/cos) and the cosine of 90 degrees is zero and you can't division by zero. 90 + 75 is out

The subtraction formula is

tan(α - β) = tanα - tanβ/(1 +tanα*tanβ)

Let α = 180

Let β = 15

And just plug in the values

tan(180 - 15) = (tan180 - tan15)/(1 + tan180*tan15)

tan(165) = (0 - (-√3 +2))/(1+ (0*(-√3 +2))

tan(165) = √3 - 2/(1+ 0)

tan(165) = √3 - 2/1

tan(165) = √3 - 2

I hope this helps

Answer 2

Here is one way to approach this problem.

First of all, we can re write tan (165°) as tan [(120°) + (45°)]

So, the addition formula we need here is

tan (A+B) = [tan (A) + tan (B)] / [1– tan (A) tan (B)] { derived from sin(A+B) /cos (A + B) }

Substituting , we have tan [(120°) + (45°)] = [tan (120°) + tan (45°)] / [1 – tan(120°) tan(45°)]

Now you can use the Unit circle or your calculator (degree mode). = [ (–√3) + 1] / [ 1–(–√3) (1)]

Now you must simplify and rationalize the denominator to get your final answer.