What is the half-angle of tangent 22.5 (degrees) ?

Find tan (22.5)

Answer: #-1 + sqrt2#

Explanation:

Call tan (22.5) = tan t - - > tan 2t = tan 45 = 1

Use trig identity: # tan 2t = (2tan t) / (1 - tan^2 t) # (1)

#tan 2t = 1 = (2tan t) / (1 - tan^2 t) # - - >

--> #tan^2 t + 2 (tan t) - 1 = 0#

Solve this quadratic equation for tan t.

#D = d^2 = b^2 - 4ac = 4 + 4 = 8# - - > #d = + - 2sqrt2#

There are 2 real roots:

tan t = - b/2a + - d/2a = - 2/1 + 2sqrt2/2 = - 1 + - sqrt2

Answer:

#tan t = tan (22.5) = - 1 + - sqrt2#

Since tan 22.5 is positive, then take the positive answer:

tan (22.5) = - 1 + sqrt2

Since 22.522.5 is not an angle where the values of the six trigonometric functions are known, try using half-angle identities. 22.522.5 is not an exact angleFirst, rewrite the angle as the product of 12 12 and an angle where the values of the six trigonometric functions are known. In this case, 22.522.5 can be rewritten as (12) ⋅45 12 ⋅45. tan ((12) ⋅45) tan 12 ⋅45 Use the half-angle identity for tangent to simplify the expression. The formula states that ta n (θ2) = sin (θ) 1 + cos (θ) ta n θ2 = sinθ 1 + cosθ. sin (45) 1 + cos (45) sin45 1 + cos45 Simplify the result.

√2 - 1