Half-Angle trig identities?

Question

i'm trying to find tan(75) using the half-angle identity for tangent, but i'm not quite sure which reference angle to use for it being i don't what sin, cos, tan, are of 75.

sorry, that should say *since i don't know know what sin, cos, tan are of 75*

Answer 1

Half-Angle Trig Identities:
sin (θ/2) = √[(1 - cos θ)/2]
cos (θ/2) = √[(1 + cos θ)/2]
tan (θ/2) = (1 - cos θ)/sin θ = sin θ/(1 + cos θ)

If we take 75° = 150°/2, then θ = 150°. Now we must know what sin 150° and cos 150° is.

150° is in Quadrant II, so sin (or y) is positive and cos (or x) is negative. The reference angle is 180° - 150° or 30°. We use the basic facts that sin 30° = 1/2 and cos 30° = √3/2

sin 150° = sin 30° = 1/2
cos 150° = - cos 30° = -√3/2

Thus, we can now use the half angle identities to evaluate tan 75°.


We begin:
tan 75°

We transform the angle into a half of another angle (w/c is 150°)
tan 75° = tan (150°/2)

We substitute either formula for tangent of half an angle.
tan 75° = (1 - cos 150°)/sin 150° or = sin 150°/(1 + cos 150°)

Now we substitute the values for cos 150° and sin 150°.
tan 75° = [1 - (-√3/2)] / (1/2) or = (1/2)/[1 + (-√3/2)]

We simplify the grouping symbols
tan 75° = (1 + √3/2)/(1/2) or = (1/2)/(1 - √3/2)

Multiply 2/2 to simplify the complex fraction
tan 75° = (2 + √3)/1 or = 1/(2 - √3)

On the second formula, rationalize the denominator by multiplying (2 + √3)/(2 + √3)
tan 75° = (2 + √3)/1 or = (2 + √3)/(2² - √3²)

Once again simplify (including the first formula)
tan 75° = 2 + √3 or = (2 + √3)/(4 - 3)

Some more simplification:
tan 75° = 2 + √3 or = (2 + √3)/(1)

Indeed, the value is:
tan 75° = 2 + √3 ≈ 3.732050808........

^_^

Answer 2

This can be solved in 2 ways

one if x = 150 degrees

sin x = 1/2
cos x = -sqrt(3)/2
1/2x = 75 degrees

tan x/2 = (1-cos x)/sin x
using the above = tan x/2 = 2(1+sqrt(3)/2) = sqrt(3) +2

but if you are not particlar about 1/2 angle

tan 75* = tan(45*+30*) = (tan 45+tan30)/(1-tan 45 tan 30)
tan 45 =1
tan 30 = 1/sqrt(3)
so tan 75 = (1+1/sqrt(3))/(1-sqrt(3)
= (sqrt(3)+1)/sqrt(3)(1-sqrt(3)
by simplifying this you can get the result

Answer 3

I would start out with the identity tan(x) = 1/tan(90-x). So tan(75) = 1/tan(90-75) = 1/tan(15) = 1/tan(30/2) = sin(30)/(1-cos(30)) =(1/2)/(1-sqrt(3)/2) = 1/(2-sqrt(3))

Answer 4

tan 150* = tan (180*-30*) = -tan 30* = -1/sqrt(3)
let 2x = 150, so x=75.
Now use the identity tan 2x = 2*tan x /(1 - tan^2 x) to solve for tan x.
Remember that since 75* lies in the first quadrant, tan75* is positive.