Question

suppose theta satisfies the equation sin theta=1/tan theta. find the value of cos theta where theta lies in the first quadrant .

Answer 1

$\cos theta = \frac{1.3}{2}$

Answer 2

Therefore, $cos\theta$ value is $0.618$

Step-by-step explanation:

Given;

$sin\theta=\frac{1}{tan\theta}$ then $cos\theta=?$ where $\theta$ lies in the first quadrant.

∴$sin\theta=\frac{1}{tan\theta}$

⇒$sin\theta=\frac{1}{\frac{sin\theta}{cos\theta} }$  (where $tan\theta=\frac{sin\theta}{cos\theta}$)

⇒$sin\theta=\frac{cos\theta}{sin\theta}$

⇒$sin^{2} \theta=cos\theta$

⇒$(1-cos^{2} \theta)=cos\theta$ (where $sin^{2} \theta+cos^{2} \theta=1$)

⇒$cos^{2}\theta+cos\theta=1$

⇒$cos^{2}\theta+2\times \frac{1}{2}\times cos\theta+\frac{1}{4} =1+\frac{1}{4}$ (by adding $\frac{1}{4}$ on both sides)

⇒$(cos\theta +\frac{1}{2})^{2} =\frac{5}{4}$

⇒$cos\theta +\frac{1}{2} =\pm\frac{\sqrt{5}}{2}$

⇒$cos\theta=-\frac{1}{2} \pm\frac{\sqrt{5}}{2}$

∵ In the first quadrant  $cos\theta$ is positive.

So,

$cos\theta=-\frac{1}{2} +\frac{\sqrt{5}}{2}$

⇒$cos\theta=\frac{-1+\sqrt{5} }{2}$

∴ $cos\theta=0.618$

∴ $cos\theta$ value is $0.618$