Question

2sin37,5•cos37,5•cos75

Answer 1

Объяснение:

$2*sin37,5^0*cos37,5^0*cos75^0=sin(2*37,5^0)*cos75^0=sin75^0*cos75^0=\\=\frac{2*sin75^0*cos75^0}{2} =\frac{sin(2*75^0)}{2}=\frac{sin150^0}{2}=\frac{\frac{1}{2} }{2}=\frac{1}{4} .$

Answer 2

To calculate the value of 2sin(37.5) * cos(37.5) * cos(75), you can use trigonometric identities and the angle sum formula for sine.

1. Start with the angle sum formula for sine:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

2. In this case, A = 37.5 degrees and B = 75 degrees.

3. Plug in these values:

sin(37.5 + 75) = sin(112.5)

4. Use the sum-to-product identities for sine:

sin(112.5) = sin(135 - 22.5)

5. Now, use the subtraction formula for sine:

sin(135 - 22.5) = sin(135)cos(22.5) - cos(135)sin(22.5)

6. Calculate the values of sin(135) and cos(135):

sin(135) = -√2/2 cos(135) = -√2/2

7. Plug these values into the equation:

sin(112.5) = (-√2/2)cos(22.5) - (-√2/2)sin(22.5)

8. Now, calculate sin(22.5) and cos(22.5):

sin(22.5) = √(1 - cos^2(22.5)) cos(22.5) = √(1 - sin^2(22.5))

9. Use these values in the equation:

sin(112.5) = (-√2/2)√(1 - cos^2(22.5)) + (√2/2)√(1 - sin^2(22.5))

10. Simplify further:

sin(112.5) = (-√2/2)√(1 - (1 - sin^2(22.5))) + (√2/2)√(1 - sin^2(22.5))

11. Simplify the expressions inside the square roots:

sin(112.5) = (-√2/2)√(sin^2(22.5)) + (√2/2)√(1 - sin^2(22.5))

12. Combine the terms:

sin(112.5) = (√2/2)sin(22.5) + (√2/2)√(1 - sin^2(22.5))

13. Substitute sin(22.5) back in using the half-angle formula for sine:

sin(22.5) = √((1 - cos(45))/2) sin(22.5) = √((1 - √2/2)/2)

14. Now, substitute this value back into the equation:

sin(112.5) = (√2/2)√(√2/2) + (√2/2)√(1 - √((1 - √2/2)/2)^2)

15. Continue to simplify:

sin(112.5) = (√2/2)(√2/2) + (√2/2)√(1 - ((1 - √2/2)/2))

16. Calculate the final result:

sin(112.5) = (2/4) + (√2/2)√(1 - ((1 - √2/2)/2))

17. Simplify the square root term:

sin(112.5) = 1/2 + (√2/2)√(2 - (1 - √2/2))

18. Simplify further:

sin(112.5) = 1/2 + (√2/2)√(3 - √2/2)

So, the value of 2sin(37.5) * cos(37.5) * cos(75) is approximately:

2sin(37.5) * cos(37.5) * cos(75) ≈ 1/2 + (√2/2)√(3 - √2/2)