MTH1010_09_02_3

Question

MTH1010_09_02_3

BlackTom AI · Accepted Answer

We start by identifying the given information and the quadrant: sin(θ) = 4/5 and 0 < θ < π/2, which places θ in the first quadrant where all trigonometric functions are positive.
Next, recall the Pythagorean identity sin^2(θ) + cos^2(θ) = 1. Since sin(θ) = 4/5, we have sin^2(θ) = (4/5)^2 = 16/25.
Therefore, cos^2(θ) = 1 − sin^2(θ) = 1 − 16/25 = 9/25.
Taking the square root gives cos(θ) = ±√(9/25) = ±3/5. However, because θ is in the first quadrant, cos(θ) must be positive, so cos(θ) = +3/5.
Now compare with the provided options:
- Option (a) 1/5 is not equal to 3/5, so it’s incorrect.
- Option (b) 3/5 exactly matches the computed value, and aligns with the quadrant sign.
- Option (c) −1/5 is negative and does not match the positive cosine in QI, so it’s incorrect.
- Option (d) −3/5 is also negative, thus inconsistent with θ in the first quadrant.
Thus the value that satisfies both the identity and the quadrant condition is 3/5, corresponding to the correct cosine value for the given sin and angle range.

MTH1010_09_02_3

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