Answer:
To find the value of sin θ given that sec θ = 7/2 and tan θ > 0, we can use the following trigonometric identities:
1. sec θ = 1/cos θ
2. tan θ = sin θ / cos θ
First, let's find the value of cos θ using the information provided:
sec θ = 7/2
Since sec θ = 1/cos θ, we can write:
1/cos θ = 7/2
Now, cross-multiply:
2 = 7cos θ
Now, solve for cos θ:
cos θ = 2/7
Next, we can use the fact that tan θ > 0 to determine the sign of sin θ. In the context of the unit circle, this means that θ must be in the first quadrant or the third quadrant, where sin θ is positive.
Now, we have cos θ = 2/7, and we can use the Pythagorean identity for sine:
sin θ = √(1 - cos² θ)
sin θ = √(1 - (2/7)²)
sin θ = √(1 - 4/49)
sin θ = √(45/49)
sin θ = √(9/9) * √(5/7)
sin θ = (3/3) * (√5/√7)
sin θ = (√5/√7)
Now, rationalize the denominator:
sin θ = (√5/√7) * (√7/√7)
sin θ = (√5√7)/(√7√7)
sin θ = (√35)/7
So, the value of sin θ is sin θ = (√35)/7, which is approximately equal to 0.748.
Therefore, the correct answer is not listed among the provided options.
Explanation: