Question

Use the Pythagorean identity sin^2 (theta) + cos^2 (theta) =1, where theta is any real number, to find: cos (theta), given sin (theta) = 5/13, for pi/2 less than theta less than pi.

Answer 1

just plug-in sin (theta) to identity and solve directly for cos (theta)....

Answer 2

\[\cos^2\theta+\left(\frac{5}{13}\right)^2=1\]\[\cos\theta=\pm\sqrt{1-\frac{25}{169}}=\pm\sqrt{\frac{169-25}{169}}=\pm\sqrt{\frac{144}{169}}=\pm\frac{12}{13}\]if \(\cos\theta=+12/13\) ...\[\theta=\cos^{-1}\frac{12}{13}=22.62^\circ\]if \(\cos\theta=-12/13\) ...\[\theta=\cos^{-1}\frac{-12}{13}=157.38^\circ\]the condition for (theta):\[\pi/2<\theta<\pi\]therefore\[\cos\theta=-12/13\]since this is where the \(\theta=157.38^\circ\) satisfy the condition...

Answer 3

Oh! I get it! thank you!