Final answer:
With cos(θ) = ½ in the fourth quadrant, we find sin(θ) = -√3/2, tan(θ) = -√3, csc(θ) = -2/√3, sec(θ) = 2, and cot(θ) = -1/√3 using Pythagoras' theorem and the property that sine is negative in the fourth quadrant.
Step-by-step explanation:
If the cosine of an angle θ is known to be ½ and the angle lies in the fourth quadrant, we know that the cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle. In the fourth quadrant, the cosine is positive and sine is negative. As cos(θ) = ½, we can represent this as a right triangle where the adjacent side (Ax) is 1 and the hypotenuse (A) is 2.
To find the length of the opposite side (Ay), we can use Pythagoras' theorem: A² = Ax² + Ay², which gives us Ay = √(A² - Ax²) = √(2² - 1²) = √(4 - 1) = √3.
Since θ is in the fourth quadrant, its sine value (Ay/A) will be negative, so sin(θ) = -√3/2. The tangent (Ay/Ax) will also be negative: tan(θ) = -√3. For the reciprocal trigonometric functions, we have cosecant (csc(θ) = 1/sin(θ) = -2/√3), secant (sec(θ) = 1/cos(θ) = 2), and cotangent (cot(θ) = 1/tan(θ) = -1/√3).