Find the exact value of cosθ if sinθ=-2/3, and θis in standard position with its terminal side in Quadrant IV.
cos is + in quadrant IV
2^2+x^2 = 3^2
x^2 = 5
x = sqrt(5)
so
sqrt(5)/3
Thankyouu
To find the exact value of cosθ given that sinθ = -2/3 and θ is in standard position with its terminal side in Quadrant IV, we can use the Pythagorean identity and the definition of cosine.
First, let's draw a right triangle in Quadrant IV:
```
|
|
|____
θ
```
Since sinθ = -2/3, we know that the side opposite angle θ is -2 and the hypotenuse is 3. Using the Pythagorean theorem, we can find the length of the adjacent side:
a^2 + b^2 = c^2
b^2 = c^2 - a^2
b^2 = 3^2 - (-2)^2
b^2 = 9 - 4
b^2 = 5
Taking the square root of both sides, we get:
b = √5
Now, we can use the definition of cosine to find the exact value of cosθ:
cosθ = adjacent / hypotenuse
cosθ = √5 / 3
Therefore, the exact value of cosθ is √5 / 3.