Prove that is an increasing function of θ in .
The given function is
y = 4 sin θ ( 2 + cos θ ) − θ .
Differentiate the function with respect to θ .
d y d θ = d d θ ( 4 sin θ ( 2 + cos θ ) − θ ) = d d θ ( 4 sin θ ( 2 + cos θ ) ) − d d θ ( θ ) = ( 2 + cos θ ) ( 4 cos θ ) − 4 sin θ ( − sin θ ) ( 2 + cos θ ) 2 − 1 = 8 cos θ + 4 cos 2 θ + 4 sin 2 θ ( 2 + cos θ ) 2 − 1
Further simplify the above expression.
d y d θ = 8 cos θ + 4 ( cos 2 θ + sin 2 θ ) ( 2 + cos θ ) 2 − 1 = 8 cos θ + 4 ( 2 + cos θ ) 2 − 1 (1)
Substitute d y d θ = 0
8 cos θ + 4 ( 2 + cos θ ) 2 − 1 = 0 8 cos θ + 4 ( 2 + cos θ ) 2 = 1 8 cos θ + 4 = 4 + cos 2 θ + 4 cos θ cos 2 θ + 4 cos θ − 8 cos θ = 0
Further simplify the above expression.
cos 2 θ − 4 cos θ = 0 cos θ ( cos θ − 4 ) = 0 cos θ = 0 or cos θ = 4
The value of cos θ cannot be equal to 4.
cos θ ≠ 4 cos θ = 0 cos θ = cos π 2 θ = π 2
Further simplify the equation (1).
d y d θ = 8 cos θ + 4 − 4 − cos 2 θ − 4 cos θ ( 2 + cos θ ) 2 = 4 cos θ − cos 2 θ ( 2 + cos θ ) 2 = cos θ ( 4 − cos θ ) ( 2 + cos θ ) 2
In the interval 0 < θ < π 2
( 4 − cos θ ) > 0 d y d θ > 0 cos θ ( 4 − cos θ ) ( 2 + cos θ ) 2 > 0
Thus, the function y = 4 sin θ ( 2 + cos θ ) − θ is strictly increasing in the interval ( 0 , π 2 ) .
The given function is
Differentiate the function with respect to
Further simplify the above expression.
Substitute
Further simplify the above expression.
The value of
Further simplify the equation (1).
In the interval
Thus, the function
