If sin = 2/3 and is a second quadrant angle, determine the exact value of sec^2

could be
a. 5/9
b. root5/3
c. 9/4
d. 9/5

Please help.......

If sin = 2/3, the side opposite from the angle is 2 and the hypotenuse is 3, since sin is opposite over hypotenuse.

a^2 + b^2 = c^2. The hypotenuse, 3, is c, and 2 can be a. 3^2 - 2^2 is 5. Since that leaves b^2 = 5, you have to take the root of both sides, so b = root5. That means the adjacent side is root5.

Cos is adjacent over hypotenuse, so cos is root5 over 3. Sec is cos flipped, so sec is 3/root5 or 3root5/5. When you square that whole thing, you get 9 times 5 (45) divided by 25, which is the same thing as 9/5.

I hope that helped!

sin^2 = 4/9

cos^2 = 1 - sin^2 = 5/9
1/cos^2 = sec^2 = 9/5

To determine the exact value of sec^2 given that sin is equal to 2/3 and is a second quadrant angle, we can use the identity:

sec^2θ = 1 + tan^2θ

First, we need to find the value of tanθ. In second quadrant, sin is positive and cos is negative. We can view sinθ = 2/3 as opposite/hypotenuse and use the Pythagorean theorem to find the missing side:

cosθ = -√(1 - sin^2θ)
= -√(1 - (2/3)^2)
= -√(1 - 4/9)
= -√(5/9)
= -√5 / 3

Now, we can find tanθ by dividing sinθ by cosθ:

tanθ = sinθ / cosθ
= (2/3) / (-√5 / 3)
= -2√5 / 5

Finally, we can substitute tanθ into the formula for sec^2θ:

sec^2θ = 1 + tan^2θ
= 1 + (-2√5 / 5)^2
= 1 + (4/5)
= 9/5

Therefore, the exact value of sec^2θ is 9/5. Among the answer choices provided, the correct option is (d) 9/5.