Question

X equals to 4 cos theta minus 5 sin theta, y equals to 4 sin theta + 5 cos theta​

Answer 1

Complete question:

Eliminate theta  from the following: x=4cos theta-5 sin theta, y=4 sin theta +5 cos theta​

Answer:

x² + y² = 6²  

Step-by-step explanation:

we are given that

x = 4cos(Ф) - 5 sin(Ф)   .....(1)

y = 4sin(Ф) + 5cos(Ф)   .....(2)

Taking square on both sides of both equations we get

x² = 4²cos²(Ф) + 5² sin²(Ф) - 2(4)(5)sin(Ф)cos(Ф)  ....(3)

y² = 4²sin²(Ф) + 5²cos²(Ф) + 2(4)(5)sin(Ф)cos(Ф)    ....(4)

Adding equation (3) and equation (4) we get

x² + y² = 4²cos²(Ф) + 5² sin²(Ф) + 4²sin²(Ф) + 5²cos²(Ф)

            = 4²cos²(Ф) + 4²sin²(Ф) + 5² sin²(Ф) + 5²cos²(Ф)

            = 4²( cos²(Ф) + sin²(Ф) ) + 5²( cos²(Ф) + sin²(Ф) )

            =  4²( 1 ) + 5²( 1 )               ∵ cos²(Ф) + sin²(Ф) = 1

            = 16 + 25

            = 36

            = 6²

Hence we the new equation in which angle is not present.

x² + y² = 6²