((4sin(atan(4/3))-4sin(0)))-(-3cos(atan(4/3))-(-3cos(0)))+((-3cos(pi/2)-(-3cos(atan(4/3))))-(4sin(pi/2)-4sin(atan(4/3))))+((-3cos(4pi/5)-(-3cos(pi/2)))-(4sin(4pi/5)-4sin(pi/2)))
To simplify this expression, follow these steps:
Step 1: Calculate the values of the trigonometric functions:
- sin(0) = 0
- cos(0) = 1
- sin(pi/2) = 1
- cos(pi/2) = 0
- sin(atan(4/3)): First, find the value of atan(4/3) using a calculator. Let's say atan(4/3) = A. Then, sin(A) will be sin(atan(4/3)).
- cos(atan(4/3)): First, find the value of atan(4/3) using a calculator. Let's say atan(4/3) = A. Then, cos(A) will be cos(atan(4/3)).
- sin(4pi/5): Use a calculator to find sin(4pi/5).
- cos(4pi/5): Use a calculator to find cos(4pi/5).
Step 2: Substitute the calculated values into the expression.
((4*sin(atan(4/3)) - 4*sin(0))) - (-3*cos(atan(4/3)) - (-3*cos(0))) + ((-3*cos(pi/2) - (-3*cos(atan(4/3)))) - (4*sin(pi/2) - 4*sin(atan(4/3)))) + ((-3*cos(4pi/5) - (-3*cos(pi/2))) - (4*sin(4pi/5) - 4*sin(pi/2)))
= ((4*sin(A) - 4*0)) - (-3*cos(A) - (-3*1)) + ((-3*0 - (-3*cos(A))) - (4*1 - 4*sin(A))) + ((-3*cos(4pi/5) - (-3*0)) - (4*sin(4pi/5) - 4*1))
= (4*sin(A) - 0) - (-3*cos(A) + 3) + (0 + 3*cos(A) - (0 - 4*sin(A))) + (-3*cos(4pi/5) + 0 - (4*sin(4pi/5) - 4))
= 4*sin(A) - (-3*cos(A) + 3) + (3*cos(A) + 4*sin(A)) - (-3*cos(4pi/5) - 4*sin(4pi/5) + 4)
= 4*sin(A) + 3*cos(A) - 3 + 3*cos(A) + 4*sin(A) - 3*cos(4pi/5) - 4*sin(4pi/5) + 4
= 8*sin(A) + 6*cos(A) - 3 - 3*cos(4pi/5) - 4*sin(4pi/5) + 4
Now, you can calculate the values of sin(A), cos(A), sin(4pi/5), and cos(4pi/5) using a calculator and substitute them back into the expression to get the final numerical result.