Solve: 5sin theta = 6 cos^2theta -6 for theta in [0, 2pi]?

2 Answers
Jan 4, 2018

theta = 0, pi, approx 4.127, approx 5.298,2pi for theta in [0,2pi]

Explanation:

5sin theta = 6 cos^2theta -6

5sin theta = 6(cos^2theta -1)

5sin theta = -6sin^2theta

sin theta(5+6sin theta) =0

sin theta = 0 or sin theta = -5/6

sin theta =0 ->theta = 0, pi or 2pi in [0, 2pi]

sin theta = -5/6 -> theta = arcsin (-5/6) approx 4.127 or approx 5.298 in [0, 2pi]

Hence, theta = 0, pi, approx 4.127, approx 5.298,2pi for theta in [0,2pi]

We can see these results from the zeros of the graph of sin theta(5+6sin theta) for theta in [0,2pi] below:

graph{sinx(5+6sinx) [-1.56, 7.21, -1.892, 2.49]}

Jan 4, 2018

theta=0; pi; 4.126; 5.298; 2pi

Explanation:

5sintheta=6cos^2theta-6

5sintheta=6(cos^2theta-1)


1=sin^2x+cos^2xquad=>quadsin^2x=1-cos^2x


5sintheta=-6(1-cos^2theta)

5sintheta=-6sin^2theta

5sintheta+6sin^2theta=0

sintheta(5+6sintheta)=0

sintheta=0quad=>quadtheta_1=0quad^^quadtheta_2=piquad^^quadtheta_3=2pi


5+6sintheta_(4,5)=0
=>sintheta_(4,5)=-5/6
To find that you have to use calculator but it will tell you only one value. To find second value we need to know in which quadrant we are.

=>theta_(4)~~-0.985~~-pi/3.1
but 0<=theta<=2pi
=>theta_4~~-0.985+2pi~~color(pink)5.298~~(5pi)/3

(3pi)/2<=(5pi)/3<=2piquad=>quad 4th quadrant
That means that the last angle must be in 3rd quadrant. To calculate that we use well known trick:
For 3rd quadrant: varphi=pi+varphi_0

Applying to our case: theta_0=|-0.985|
=>theta_5=pi+0.985=color(orange)4.126

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