Question

Вычислите: cos2a-sin2a+sin3a , если a=30°.

Answer 1

Cos60-sin60+sin90= 1/2 +  $\sqrt{3} /2$ +1 = 2,37

Answer 2

To calculate the value of the expression cos(2a) - sin(2a) + sin(3a) when a = 30°, we first need to find the values of cos(2a), sin(2a), and sin(3a) for a = 30°.

Using trigonometric identities:

1. cos(2a) = cos^2(a) - sin^2(a)
2. sin(2a) = 2 * sin(a) * cos(a)
3. sin(3a) = 3 * sin(a) - 4 * sin^3(a)

Given that a = 30°, we can use the values of sin(30°) and cos(30°) that are well-known:

sin(30°) = 1/2 cos(30°) = √3/2

Let's calculate the values step by step:

1. cos(2a) = cos^2(a) - sin^2(a) cos(2 * 30°) = cos^2(30°) - sin^2(30°) cos(60°) = (√3/2)^2 - (1/2)^2 cos(60°) = 3/4 - 1/4 cos(60°) = 2/4 cos(60°) = 1/2

2. sin(2a) = 2 * sin(a) * cos(a) sin(2 * 30°) = 2 * sin(30°) * cos(30°) sin(60°) = 2 * (1/2) * (√3/2) sin(60°) = √3/2

3. sin(3a) = 3 * sin(a) - 4 * sin^3(a) sin(3 * 30°) = 3 * sin(30°) - 4 * sin^3(30°) sin(90°) = 3 * (1/2) - 4 * (1/2)^3 sin(90°) = 3/2 - 4 * (1/8) sin(90°) = 3/2 - 1/2 sin(90°) = 2/2 sin(90°) = 1

Now, we can substitute these values back into the original expression:

cos(2a) - sin(2a) + sin(3a) = 1/2 - √3/2 + 1

Now, let's simplify the expression:

1/2 - √3/2 + 1 = (1 - √3 + 2) / 2 = (3 - √3) / 2 ≈ 0.366

So, cos(2a) - sin(2a) + sin(3a) is approximately 0.366 when a = 30°.