Suppose that (x,5/13) is a point in quadrant 2 lying on the unit circle. Find x. Write the exact value, not a decimal approximation.
or, in QII, x = -12/13
Well, I'm not really good at math, but I can surely come up with a clowny response for you! Here you go:
Why did the unit circle go to therapy?
Because it had too many issues with its decimal approximations!
But fret not, I'll try my best to help you out. In quadrant 2, both the x and y coordinates are negative. Since the point lies on the unit circle, we know that the hypotenuse is equal to 1.
So, using the Pythagorean theorem, we can find the value of x by calculating:
x^2 + (5/13)^2 = 1
Let's solve this equation step by step:
x^2 + 25/169 = 1
x^2 = 1 - 25/169
x^2 = 169/169 - 25/169
x^2 = 144/169
Taking the square root of both sides gives us:
x = ±√(144/169)
Therefore, the exact values of x would be ±12/13.
I hope that helps! And remember, when it comes to math, always keep a clown-face on to make it more fun!
To find the value of x in the given quadrant-2 point on the unit circle, we can use the Pythagorean Identity for the unit circle:
sin^2(theta) + cos^2(theta) = 1
Since the given point lies on the unit circle, the x-coordinate would be equal to the cosine value and the y-coordinate would be equal to the sine value.
Given: (x, 5/13)
Now, we can set up the equation using the given information:
x = cos(theta)
5/13 = sin(theta)
To find the exact value of x, we need to solve for theta using the given sin(theta) value.
Using the Pythagorean Identity:
sin^2(theta) + cos^2(theta) = 1
(5/13)^2 + x^2 = 1
25/169 + x^2 = 1
x^2 = 1 - 25/169
x^2 = (169/169) - (25/169)
x^2 = 144/169
Taking the square root of both sides:
x = ± sqrt(144/169)
x = ± (12/13)
Since the point lies in quadrant 2, the x-coordinate would be negative. Therefore, x = -12/13.
To find the value of x, we can use the concept of trigonometric functions and the geometry of the unit circle.
In quadrant 2 of the unit circle, the x-coordinate is negative, and the y-coordinate is positive. We are given that the point lies on the unit circle, which means the distance of the point from the origin is 1.
Since the actual y-coordinate is 5/13, we need to find a common denominator to express it in terms of 13. Multiplying the numerator and denominator by 13, we have (x, 65/13).
Now, consider the right triangle formed by the point on the unit circle, the x-axis, and the line segment connecting the point to the origin. Since the distance from the origin to the point is 1, the hypotenuse of the triangle is 1.
Let's label the angle formed between the hypotenuse and the positive x-axis as θ. We can use the trigonometric function sine (sin θ) to relate the sides of the triangle:
sin θ = opposite/hypotenuse.
In this case, the opposite side is the y-coordinate (65/13), and the hypotenuse is 1:
sin θ = (65/13)/1.
Simplifying the expression, we have:
sin θ = 65/13.
To find the value of θ in quadrant 2, we can use the inverse sine function (arcsin). Applying arcsin to both sides:
θ = arcsin(65/13).
Now, using a scientific calculator, we can compute the value of arcsin(65/13). The exact value of θ will be returned as an inverse sine function.
Once we have the value of θ, we can determine the value of x by applying the cosine function (cos) to that angle, as follows:
x = cos θ.
Calculating the cosine of the angle will give us the exact value of x without any decimal approximation.
x is negative in Q 2
(n/13)^2 + (5/13)^2^2 = (13/13)^2
12=n
so
x = 12/13