Point P (a, -2/5) is located in the third quadrant on the unit circle. Determine the exact value of a. Show your work.
it's the unit circle, so x^2 + y^2 = 1
a^2 + 4/25 = 25/25
a^2 = 21/25
a = -√21/5
To determine the exact value of a for point P, we can use the equation of the unit circle:
x^2 + y^2 = 1
Since point P is located in the third quadrant, we know that the x-coordinate, a, will be negative. So, we have:
a^2 + (-2/5)^2 = 1
To solve for a, we can rearrange the equation:
a^2 + 4/25 = 1
Subtracting 4/25 from both sides:
a^2 = 1 - 4/25
a^2 = 25/25 - 4/25
a^2 = 21/25
Taking the square root of both sides:
√(a^2) = √(21/25)
a = ± √(21/25)
Since point P is in the third quadrant, the x-coordinate a will be negative. Therefore, the exact value of a is:
a = - √(21/25)
So, the exact value of a is -√21/5.
To determine the value of `a`, we need to use the information given about the point P being located in the third quadrant on the unit circle.
For a point on the unit circle, the coordinates (`x`, `y`) represent the values of `cos(theta)` and `sin(theta)` respectively, where `theta` is the angle formed between the positive x-axis and the line connecting the origin (0,0) to the point on the unit circle.
In the third quadrant, both `x` and `y` coordinates are negative. Since the y-coordinate is -2/5, we can write `y = -2/5` and `x = -sqrt(1 - y^2)`.
To solve for `a`, we will substitute `-2/5` for `y` in the expression `x = -sqrt(1 - y^2)`:
`a = -sqrt(1 - (-2/5)^2)`
Simplifying:
`a = -sqrt(1 - 4/25)`
`a = -sqrt(25/25 - 4/25)`
`a = -sqrt(21/25)`
As `a` needs to be in the third quadrant, it should be negative. Hence, the exact value of `a` is:
`a = -sqrt(21)/5`
Therefore, the exact value of `a` is `-sqrt(21)/5`.