Final answer:
To solve sec(tan⁻¹(4/7)), we can find the tangent of the angle, take its reciprocal to get the cosine, and then take the reciprocal again to find the secant. The value of sec(tan⁻¹(4/7)) is approximately 1.75.
Step-by-step explanation:
To solve sec(tan⁻¹(4/7)), we can use the definition of secant as the reciprocal of cosine and the definition of tangent as the sine divided by the cosine.
Let's start by finding the value of tan⁻¹(4/7). The angle whose tangent is 4/7 is in the first quadrant because both the numerator and the denominator are positive. Using the inverse tangent function, we can find that tan⁻¹(4/7) ≈ 30.96° or 0.54 radians.
Next, we can find the value of sec(tan⁻¹(4/7)) by taking the reciprocal of the cosine of the angle. The cosine of the angle is equal to the adjacent side divided by the hypotenuse in a right triangle with that angle. Since the tangent of the angle is 4/7, we can let the adjacent side be 4 and the hypotenuse be 7. Therefore, the cosine of the angle is 4/7. Taking the reciprocal, we get 7/4 ≈ 1.75. Hence, sec(tan⁻¹(4/7)) ≈ 1.75.