Homework SourseGive the approximate answers as decimals accurate to the nea
Give the approximate answers as decimals, accurate to the nearest 0.001. All trigonometric functions are assumed to be in radians. Each problem is worth 2 points.
| 1. | Rewrite the given function in the form acos(5x) + bsin(5x). Enter the missing coefficients as decimals accurate to within 0.001. 6cos(5x + 2.26) = |
| cos(5x) + sin(5x) | |
| 2. | Rewrite the given function in the form acos(x) + bsin(x). Enter the missing coefficients as decimals accurate to within 0.001. 3sin(x 0.99) = |
| cos(x) + sin(x) | |
| 3. | Rewrite the given function in the form Acos(7x ), where A > 0 and -/2 < < /2. Enter the missing coefficients as decimals accurate to within 0.001. 7cos(7x) 5sin(7x) = |
| cos(7x ) | |
| 4. | Let f(x) = cos(4x) + 7sin(4x). Find the maximum value f(x) attains and the point x closest to 0 where it attains its max (since the function is periodic, it attains its max every 2/4 = /2). |
| The maximum value of f(x) is = | |
| The value of x closest to 0 where this maximum is obtained is = | |
| 5. | Find all values of x with -pi<x<pi such that 8cos(x) + 2sin(x) = -5.52. Enter your answers as decimals rounded to the nearest 0.001, separating values with commas. x = |
Solution
1)6cos(5x + 2.26)
cos(A+B)=cosAcosB-sinAsinB
=6cos(5x)cos(2.26) -6sin(5x)sin(2.26)
=6cos(2.26)cos(5x) -6sin(2.26)sin(5x)
=-3.186cos(5x)-4.631sin(5x)
===============================================
2)3sin(x 0.99)
sin(A-B)=sinAcosB-cosAsinB
=3(sin(x)cos(0.99)- cos(x)sin(0.99))
=3cos(0.99)sin(x)- 3sin(0.99)cos(x)
=1.646sin(x)-2.508cos(x)
=======================================
3)7cos(7x) 5sin(7x)
=(72+52)[(7/(72+52))cos(7x) (5/(72+52))sin(7x)]
=(74) [cos(7x) (7/74) sin(7x) (5/74)]
=8.602 [cos(7x+0.620)]
=8.602 [cos(7x+0.620-2)]
=8.602 cos(7x-5.633)
