Question

If 0 < θ < $\frac{π}2 $ and tanθ = $\frac{\sqrt5}2$, then cosθ is equal to

• A. $\frac{1}2$
• B. $\frac{\sqrt3}2$
• C. $\frac{1}3$
• D. $\frac{2}3$
• E. $\frac{\sqrt5}3$

Answer 1

The Correct Option is D

Given:

\( \tan \theta = \frac{\sqrt{5}}{2} \) 

Using the identity:

\(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we assume \(\sin \theta = \sqrt{5}k\) and \(\cos \theta = 2k\) for some \(k\).

Using the Pythagorean identity:

\(\sin^2 \theta + \cos^2 \theta = 1\)

Substituting the values:

\((\sqrt{5}k)^2 + (2k)^2 = 1\)

\(5k^2 + 4k^2 = 1\)

\(9k^2 = 1\)

\(k^2 = \frac{1}{9}\)

\(k = \frac{1}{3}\) (taking the positive value as \(0 < \theta < \frac{\pi}{2}\))

Thus,

\(\cos \theta = 2k = 2 \times \frac{1}{3} = \frac{2}{3}\)

Therefore, the correct answer is:

\(\frac{2}{3}\)