How do you solve tantheta = sqrt(2)sintheta?

2 Answers
Mar 31, 2017

theta = (pm k pi uu pm pi/4 + 2kpi) for k in ZZ

Explanation:

tan(theta)=sqrt(2)sin(theta) or

tan(theta)(1-sqrt(2)cos(theta))=0

{(tan(theta)=0->theta=pm k pi),(cos(theta)=1/sqrt(2)->theta = pm pi/4 + 2kpi):}

theta = (pm k pi uu pm pi/4 + 2kpi) for k in ZZ

Mar 31, 2017

theta={pi/4, (7pi)/4}

Explanation:

Here's an alternative approach. Note that tantheta = sin theta/costheta.

sintheta/costheta = sqrt(2)sintheta

Multiply both sides by costheta.

sintheta/costheta * costheta = sqrt(2)sintheta*costheta

Now recognize that sin2theta = 2sinthetacostheta.

sintheta = sqrt(2)sinthetacostheta

Multiply the right side by 2/2
sintheta = 2sqrt(1/2)sinthetacostheta

sintheta = sqrt(1/2)sin2theta

sintheta/(2sinthetacostheta) = 1/sqrt(2)

1/(2costheta) = 1/sqrt(2)

sqrt(2) = 2costheta

sqrt(2)/2 = costheta

This is the rationalized form of 1/sqrt(2).

1/sqrt(2) = costheta

This has solutions theta = pi/4, (7pi)/4.

Hopefully this helps!