Question

If sec θ = 5/4 on the interval (0, 90°), then find tan θ.

A) tan θ = 4/5
B) tan θ = 5/4
C) tan θ = 1/5
D) tan θ = 1/4

Answer 1

The value of tan θ can be found using the relationship between sec θ and tan θ. By substituting the given value of sec θ into the equation and solving for sin θ and cos θ, we can determine the value of tan θ.

Step-by-step explanation:

To find the value of tan θ, we can use the relationship between the trigonometric functions sec θ and tan θ. Since sec θ = 5/4, we know that cos θ = 4/5. Using the identity tan θ = sin θ / cos θ, we can find sin θ as well. Let's solve for θ:

• We know that cos θ = 4/5.
• Using the Pythagorean identity sin² θ + cos² θ = 1, we can substitute the value of cos θ and solve for sin θ: sin² θ + (4/5)² = 1.
• Now, solve for sin θ: sin² θ + 16/25 = 1. Subtract 16/25 from both sides: sin² θ = 9/25.
• Take the square root of both sides: sin θ = ±3/5. Since we are working in the interval (0, 90°), the positive value of sin θ applies: sin θ = 3/5.
• Finally, substitute the values of sin θ and cos θ into the relationship tan θ = sin θ / cos θ: tan θ = (3/5) / (4/5) = 3/4.

Therefore, the correct answer is: tan θ = 3/4.