Final answer:
The value of cos θ is 1/√5, given that tan θ = √5/2. The value is found by using trigonometric identities and the Pythagorean theorem. The option (c) is correct.
Step-by-step explanation:
If tan θ = √5/2, we want to find the value of cos θ. To do this, we need to remember the identity tan θ = sin θ/cos θ and the Pythagorean theorem as it applies to trigonometry: sin2 θ + cos2 θ = 1. In a right triangle, if the opposite side is y, the adjacent side is x, and the hypotenuse is h, then tan θ = y/x, and cos θ = x/h.
To find cos θ, we assume the hypotenuse (h) is 1, because we're interested in the ratio only. This means x = cos θ and y = sin θ. Since tan θ = √5/2, we can write this as y/x = √5/2 with x being cos θ and y being sin θ. Using the Pythagorean theorem: 1 = sin2 θ + cos2 θ = (y2) + (x2). Replacing y with √5/2 * x gives us 1 = (5/4)x2 + x2. Solving this equation for x gives us x = cos θ = 1/√5, which means option (c) is correct.