Final answer:
The exact value of sec(3π/4) is found through understanding that it's the reciprocal of the cosine for a reference angle of π/4, which is 1/√2. Since 3π/4 lies in the second quadrant where cosine is negative, the exact value of sec(3π/4) is -√2.
Step-by-step explanation:
The subject of this question is Mathematics, specifically trigonometry. To find the exact value of sec(3π/4) using a triangle reference, we must first understand what secant (sec) represents in a right triangle context. For an angle θ, sec(θ) is defined as the ratio of the hypotenuse to the adjacent side (sec(θ) = 1/cos(θ)). In a unit circle, sec(θ) represents the length of the line segment from the origin to a point on the terminal side of the angle intersected by the vertical line passing through that point.
An angle of 3π/4 radians corresponds to 135 degrees, which lies in the second quadrant of the unit circle where the cosine values are negative and the secant values are also negative. Therefore, to find sec(3π/4), we can look at the corresponding reference angle of π/4 radians (45 degrees) in the first quadrant. The cosine of 45 degrees in a right triangle with equal legs is 1/√2, so the secant (which is the reciprocal) would be √2. Since we are in the second quadrant, the exact value of sec(3π/4) is -√2.