Final answer:
The exact value of sec Θ, when the point (7,6) lies on the terminal side of Θ, is found by first using the Pythagorean theorem to calculate the hypotenuse of the right triangle. Then, by taking the reciprocal of the cosine of Θ, which is the adjacent side (7) over the hypotenuse we calculated, we find the secant and simplify it to get a rational denominator. The exact value in simplest form is thus sec Θ = 7/3, which is option D.
Step-by-step explanation:
We can find the exact value of sec Θ when angle Θ is in standard position, and point (7,6) lies on the terminal side of Θ. Remember that sec Θ = 1/cos Θ, which is the reciprocal of the cosine function.
In the context of a right triangle, if (7,6) represents the x-coordinate (adjacent side) and the y-coordinate (opposite side) respectively, then we can use the Pythagorean theorem to find the hypotenuse (r) of the right triangle formed. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (r) is equal to the sum of the squares of the other two sides (x and y).
The calculation would be r = √(x² + y²) = √(7² + 6²) = √(49 + 36) = √85. Now, cos Θ = x/r = 7/√85. Therefore, sec Θ = 1/cos Θ = √85/7. Simplify the denominator to have a rational number: sec Θ = (√85/7)(√85/√85) = 85/7√85. And finally, by simplifying the radical in the denominator, we get sec Θ = 85/7√85 = 85/7(85)², resulting in sec Θ = 7/√85 × (√85/√85) = 7√85/85. To get the simplest form with a rational denominator, we divide both the numerator and the denominator by √85, which yields: sec Θ = 7/3, which corresponds to option D.