Answer:
Part 1: √53 / 7
Part 2: √53 / 2
Part 3: 7 / √53
Step-by-step explanation:
tan θ = 2/7, 0 < θ < π/2
We're given the tangent of the angle, and we're told the angle is in the first quadrant. For angles in the first quadrant, sine and cosine are both positive. Therefore, secant and cosecant will also be positive.
To find secant from tangent, use Pythagorean identity:
sin² θ + cos² θ = 1
Divide both sides by cos² θ:
tan² θ + 1 = sec² θ
Substitute and solve:
(2/7)² + 1 = sec² θ
4/49 + 1 = sec² θ
53/49 = sec² θ
sec θ = √53 / 7
Secant is the inverse of cosine, so we can now solve part 3:
cos θ = 7 / √53
To solve part 2, use the Pythagorean identity:
sin² θ + cos² θ = 1
sin² θ + (7 / √53)² = 1
sin² θ + 49/53 = 1
sin² θ = 4/53
sin θ = 2 / √53
Cosecant is the inverse of sine, so:
csc θ = √53 / 2