Question

How do you find the exact values of tan 112.5 degrees using the half angle formula?

Answer 1

`tan(112.5)=-(1+sqrt(2))`

Explanation:

`112.5=112 1/2=225/2`

NB : This angle lies in the 2nd Quadrant.

`=>tan(112.5)=tan(225/5)=sin(225/2)/cos(225/2)=-sqrt([sin(225/2)/cos(225/2)]^2)=-sqrt(sin^2(225/2)/cos^2(225/2))`

We say it's negative because the value of `tan` is always negative in the second quadrant!

Next, we use the half angle formula below :

`sin^2(x/2)=1/2(1-cosx)`

`cos^2(x/2)=1/2(1+cosx)`

`=>tan(112.5)= -sqrt(sin^2(225/2)/cos^2(225/2))= -sqrt((1/2(1-cos(225)))/(1/2(1+cos(225))))= -sqrt((1-cos(225))/(1+cos(225)))`

Notice that : `225=180+45 => cos(225)=-cos(45)`

`=>tan(112.5)=-sqrt((1-(-cos45))/(1+(-cos45)))=-sqrt((1+sqrt(2)/2)/(1-sqrt(2)/2))=sqrt((2+sqrt(2))/(2-sqrt(2)))`

Now you want to Rationalize;

`=>-sqrt(((2+sqrt(2))xx(2+sqrt(2)))/((2-sqrt(2))xx(2+sqrt(2))))= -sqrt(((2+sqrt(2))^2)/(4-2))=-(2+sqrt(2))/sqrt(2)=-(sqrt(2)xx(2+sqrt(2)))/(sqrt2xxsqrt2)=-(2sqrt2+2)/2=color(blue)(-(1+sqrt(2)))`

Answer 2

Find tan 112.5

Ans: (-1 - sqrt2)

Explanation:

Call tan 112.5 = tan t
tan 2t = tan 225 = tan (45 + 180) = tan 45 = 1
Use trig identity: `tan 2t = (2t)/(1 - t^2)` -->

`1 = (2t)/(1 - t^2)` --> `t^2 + 2t - 1 = 0`
`D = d^2 = b^2 - 4ac = 4 + 4 = 8 --> d = +- 2sqrt2`

`t = tan 112.5 = -2/2 +- (2sqrt2)/2 = - 1 +- sqrt2`

Since t = 112.5 deg is in Quadrant II, its tan is negative, then only the negative answer is accepted : (-1 - sqrt2)