Final answer:
To express sec θ in terms of sin θ in quadrant I, you can use sec θ = 1/cos θ and the Pythagorean identity to find that sec θ = 1/(√(1 - sin² θ)).
Step-by-step explanation:
To write the first trigonometric function (sec θ) in terms of the second (sin θ) for θ in quadrant I, we can use the Pythagorean identity. In any right triangle, the hypotenuse (h), the opposite side (y), and the adjacent side (x) have a specific relationship defined by the functions sine (θ) and cosine (θ), where sin θ = y/h and cos θ = x/h. Given that secant is the reciprocal of cosine, sec θ = 1/cos θ.
Using the Pythagorean Theorem, we also know that x² + y² = h². Dividing each term by h², we get (x/h)² + (y/h)² = 1, which simplifies to cos² θ + sin² θ = 1. From this identity, we can solve for cos θ as √(1 - sin² θ), but since θ is in quadrant I, where all trigonometric functions are positive, we do not require the absolute value and can directly take the positive square root, thus cos θ = √(1 - sin² θ).
To express sec θ in terms of sin θ, we substitute this expression for cos θ into the formula for sec θ:
sec θ = 1/(√(1 - sin² θ))