How can sin(7π/12 ) can be evaluated using compound angle formulas in two different ways?
sin(2 * 7π/6)
cos(π/2 - 7π/12) = cos(1/2 * π/6)
and of course the obvious one:
sin(7π/12 )
= sin(π/3 + π/4)
= sin π/3 cos π/4 + cos π/3 sin π/4
= (√3/2)(√2/2) + (1/2)(√2/2)
= (√6 + √2)/4
To evaluate sin(7π/12) using compound angle formulas, we can utilize two different approaches.
Approach 1: Double Angle Formula
Step 1: Start with the angle 7π/12.
Step 2: Rewrite the angle as the sum of two angles: 7π/12 = π/4 + π/6.
Step 3: Apply the double angle formula sin(θ + φ) = sin(θ)cos(φ) + cos(θ)sin(φ).
Step 4: Substitute the values and evaluate.
sin(7π/12) = sin(π/4 + π/6)
= sin(π/4)cos(π/6) + cos(π/4)sin(π/6)
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2)/4
Approach 2: Half Angle Formula
Step 1: Start with the angle 7π/12.
Step 2: Rewrite the angle as the sum of two angles: 7π/12 = π/3 + π/12.
Step 3: Apply the half angle formula sin(θ/2) = ±√[(1 - cos(θ))/2].
Step 4: Calculate the value of cos(π/12) using the exact trigonometric ratios or a calculator.
cos(π/12) ≈ 0.9659 (rounded to 4 decimal places).
Step 5: Substitute the value of cos(π/12) into the half angle formula.
sin(7π/12) = sin(π/3 + π/12)
= sin((π/3)/2) (since π/12 = (π/3)/2)
= ±√[(1 - cos(π/12))/2]
= ±√[(1 - 0.9659)/2]
= ±√[0.01705/2]
≈ ±√0.00853
≈ ±0.0923 (rounded to 4 decimal places)
In summary, sin(7π/12) can be evaluated as (√6 + √2)/4 using the double angle formula or approximately ±0.0923 using the half angle formula, depending on the choice of the positive or negative sign.