Express the equation 5.5 Cost + 7.8 Sin t for values 0<t<2π rad

Question

Solve for t for values in the range 0< t < 2 pie rad: 5.5 Cost + 7.8 Sin t = 4.5

Answer 1

Equate 5.5cos(t)+7.8sin(t)=4.5 to asin(x)cos(t)+acos(x)sin(t)=4.5.

asin(x)=5.5, acos(x)=7.8⇒tan(x)=5.5/7.8 and a²=5.5²+7.8².

a=9.5441 approx. x=arctan(5.5/7.8)=0.6142 radians approx.

asin(x)cos(t)+acos(x)sin(t)=asin(x+t)=9.5441sin(t+0.6142)=4.5.

sin(t+0.6142)=4.5/9.5441=0.4715 approx. t+0.6142=arcsin(0.4715)=0.4910 radians approx.

Another solution for t+0.6142 = π-0.4910=2.6506 radians approx.

So  t+0.6142=0.4910 or 2.6506, making t=-0.1232 or 2.0364 radians.

-0.1232 is 2π-0.1232=6.1600 radians.

SOLUTION t=2.0364 or 6.1600 radians.