Can someone help me answer this question? 'find the exact value of Cos theta, given sin theta = 5/9 and cos theta < 0

3 Answers
Jun 27, 2018

I got 0.8310.831

Explanation:

Have a look:
enter image source here

Jun 27, 2018

cos(theta)= -0.83147941928309808568527749607034cos(θ)=0.83147941928309808568527749607034

= -0.8315 (4dp)

Explanation:

Find thetaθ (Note: there are 2 possible angles for sin(theta) = 5/9 sin(θ)=59)
Sin^-1(5/9) = 33.748988595888587817404332744458 (degrees)sin1(59)=33.748988595888587817404332744458(degrees)

The sin will also have the value (5/9)(59) at

180-33.74898859588858781740433274445818033.748988595888587817404332744458

=146.25101140411141218259566725554 (degrees)=146.25101140411141218259566725554(degrees)

If you are not sure which of these angles will have cos < 1, try both.
(note that the numbers for both angles are actually the same but one is positive and the other is negative)

cos(33.748988595888587817404332744458)cos(33.748988595888587817404332744458)
=0.83147941928309808568527749607034=0.83147941928309808568527749607034

cos(146.25101140411141218259566725554)cos(146.25101140411141218259566725554)
=-0.83147941928309808568527749607034=0.83147941928309808568527749607034

(so thetaθ is the 146.25101.... value)

Jun 27, 2018

costheta=-sqrt56/9cosθ=569

Explanation:

"using the "color(blue)"trigonometric identity"using the trigonometric identity

•color(white)(x)sin^2theta+cos^2theta=1xsin2θ+cos2θ=1

rArrcostheta=+-sqrt(1-sin^2theta)cosθ=±1sin2θ

"given "sintheta=5/9" and "costheta<0given sinθ=59 and cosθ<0

"then "theta" is in the second quadrant"then θ is in the second quadrant

costheta=-sqrt(1-(5/9)^2)cosθ=1(59)2

color(white)(costheta)=-sqrt(1-25/81)cosθ=12581

color(white)(costheta)=-sqrt(56/81)=-sqrt56/9cosθ=5681=569