Question

Cos510 - sin480+ cos840 + sin1230

Answer 1

Учитывая, что 360 - полный период синуса и косинуса и с учётом формул сведения, допустимы следующие преобразования:
cos510 = cos(360+150) = cos(150) = cos(180-30) = -cos30
sin480 = sin(360+120) = sin(120) = sin (90+30) = cos30
cos840 = cos(840 - 2*360) = cos120 = cos(90+30) = -sin30
sin1230 = sin(1230 - 3*360) = sin150 = sin(180-30) = sin30
-cos30 - cos30 -sin30 + sin 30 = -2cos30 = -корень с 3

Answer 2

To solve this expression, we can use trigonometric identities and values of common angles.

First, we need to convert all angles to be within the range of 0 to 360 degrees or 0 to 2π radians.

• 510 degrees is equivalent to (510 - 360) = 150 degrees or 5π/6 radians
• 480 degrees is equivalent to (480 - 360) = 120 degrees or 2π/3 radians
• 840 degrees is equivalent to (840 - 720) = 120 degrees or 2π/3 radians
• 1230 degrees is equivalent to (1230 - 1080) = 150 degrees or 5π/6 radians

Now, we can substitute the values into the expression and simplify:

cos510 - sin480 + cos840 + sin1230

= cos(5π/6) - sin(2π/3) + cos(2π/3) + sin(5π/6)

= (sqrt(3)/2)/2 - sqrt(3)/2 + 1/2 + (sqrt(3)/2)/2 (using the values of cos and sin for 5π/6 and 2π/3)

= 1/2

Therefore, the value of the expression is 1/2.