Evaluate 2sin15° cos15°
2 sin x cos x = sin 2x = sin 30 = 1/2
Find the value of 2 sin15°cos15°
To evaluate the expression 2sin15° cos15°, you can use the trigonometric identity for the product of sine and cosine. The identity states that sin2θ = 2sinθcosθ.
In this case, θ is 15°.
So, sin2(15°) = 2sin(15°)cos(15°).
Now, let's calculate sin2(15°). Since we know that sin30° = 1/2, we can use the half-angle identity for sine, which states that sin(θ/2) = sqrt((1 - cosθ) / 2).
sin(15°/2) = sqrt((1 - cos15°) / 2) = sqrt((1 - cos15°) / 2).
Since sin2(15°) = 2sin(15°)cos(15°), we can now substitute the value of sin(15°) and solve for cos(15°).
2sin(15°)cos(15°) = 2 * sqrt((1 - cos15°) / 2) * cos(15°).
Simplifying further, we get:
2sin(15°)cos(15°) = sqrt((1 - cos15°) / 2) * 2 * cos(15°).
Now, we need to evaluate the value of cos(15°) using a calculator.
cos(15°) ≈ 0.96592582628907.
Substituting this value back into our expression, we get:
2sin(15°)cos(15°) = sqrt((1 - cos15°) / 2) * 2 * 0.96592582628907.
After calculating the expression, we find that
2sin(15°)cos(15°) ≈ 0.25881904510252.
Therefore, 2sin15° cos15° is approximately equal to 0.2588.