Question

If θ is an angle in standard position that terminates in Quadrant IV such that cosθ = 3/5, then cosθ/2 = _____.

a) - (√5)/5
b) - (2√2)/5
c) - (2√5)/5
d) - (4√5)/5

Answer 1

To find cos(θ/2) given cosθ = 3/5 for θ in the fourth quadrant, we use the half-angle formula. Since the angle θ/2 would lie in the second quadrant, the cosine will be negative. The correct answer is (c) - (2√5)/5 after simplifying the expression.

Step-by-step explanation:

If θ is an angle in standard position that terminates in Quadrant IV such that cosθ = 3/5, we must find cosθ/2. In this case, we can employ the half-angle formula for cosine, which is cos(θ/2) = ±√((1+cosθ)/2). Since we are dealing with an angle in the fourth quadrant, the value of cos(θ/2) will be positive because cosine is positive in the fourth quadrant. Plugging our given value into the formula, we get cos(θ/2) = ±√((1+(3/5))/2) = ±√((8/5)/2) = ±√(4/5) = ±(2/√5).

However, since we are taking the half of an angle in the fourth quadrant, the resulting angle will be in the second quadrant where cosine is negative. Therefore, the correct answer with the negative sign must be chosen. The simplified form of -2/√5, rationalizing the denominator, gives us -2√5/5, which corresponds to the option (c) - (2√5)/5.