Final answer:
To solve sec θ + tan θ = 3, we use trigonometric identities to express everything in terms of cosine and secant. After simplifying, we find cos θ = 3/5, which gives us sec θ = 5/3 as the correct answer.
Step-by-step explanation:
To solve the equation sec θ + tan θ = 3 for the value of sec θ, we can use the identity tan θ = sin θ / cos θ and the definition of secant as sec θ = 1 / cos θ. We can write the given equation in terms of cosine:
1 / cos θ + (sin θ / cos θ) = 3
Combining the terms over a common denominator gives us:
(1 + sin θ) / cos θ = 3
To solve for cos θ, we can cross-multiply:
1 + sin θ = 3 cos θ
Since cos^2 θ + sin^2 θ = 1, we can replace sin θ with √(1 - cos^2 θ):
1 + √(1 - cos^2 θ) = 3 cos θ
After squaring both sides and solving for cos θ, we find that cos θ = 3/5, which means that sec θ = 5/3.
Therefore, the correct answer is B. 5/3.