How do you find the value of tan[(12)cos1(23)]?

1 Answer
Oct 4, 2015

tanarccos(23)2=55

Explanation:

From the pythagorean identity we have that

sin2x+cos2x=1

Dividing both sides by cos2x we have

tan2x+1=1cos2x

Which means that, if we isolate the tangent we have

tan2x=1cos2x1

So for x=arccos(23)2 we have

tan2arccos(23)2=1cos2(arccos(23)2)1

cos2arccos(23)2 can be calculated using the half angle formula, so we know that

cos2arccos(23)2=1+cos(arccos(23))2

Since cos(arccos(x))=x we can say that

cos2arccos(23)2=1+232=532=5312=56

Putting that back on the formula

tan2arccos(23)2=1561=651=655=15

Taking the root

tanarccos(23)2=±15

Knowing that during the range of the arccosine, the tangent is only negative if the cosine's also negative we have that

tanarccos(23)2=15=55