Using a half-angle identity to find the exact value of tan 5pi/12, the result is: 2 + √3. See solution/explanation below.
What is half-Angle Identify?
Half-angle identities are a series of equations that can be used to convert trigonometric values of unknown angles to more recognizable ones.
Step by Step Solution is?
Step 1 - Expand the given equation
5π/12 = 3π/12 + 2π/12 which in turn = π/4 + π/6
Step 2 Using the rule of Cos we state that
Cos (a + b) = Cos a Cost b - Sina Sinb, hence
a = π/4 b = π/6
Therefore,
Cos (5π/12) = Cos((π/4) + (π/6))
= Cos(π/4) Cos(π/6) - Sin(π/4) Sin (π/6)
= (√2/2 x √3/2) - (√2/2 x (1/2)
Cos (5π/12) thus = (√6 - √2)/4
Step 3 - Reference the Sine Rule and make substitutions
Recall that
Sin (a + b) = Sin a Cos b + Cos a Sin b, hence
Sin (5π/12) = Sin (π/4 + π/6)
We expand the same to get:
Sin (π/4) Cos (π/6) + Cos (π/4) Sin (π/6)
= (√2/2 x √3/2) + (√2/2 x 1/2)
⇒ Cos (5π/12) = (√6 + √2)/4
Therefore,
Tan (5π/12) = (Sin 5π/12)/Cos 5π/12) = (√6+√2)/(√6 - √2)
= (8 + 2√12)/4
= 2 + √3
Learn more about half-angle identity at:
https://brainly.com/question/2511830
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