If point G(a,b) lies on the terminal side of angle theta in standard position, what is the value of sin(theta) if point G also lies on the unit circle?
b/1 = b
To determine the value of sin(theta) if point G(a,b) lies on the terminal side of angle theta in standard position and also lies on the unit circle, we can use the Pythagorean identity in trigonometry.
The Pythagorean identity states that for any angle theta in a right triangle, the square of the sine of theta plus the square of the cosine of theta is equal to 1. In other words:
sin^2(theta) + cos^2(theta) = 1
Since point G(a,b) lies on the unit circle, the distance between the origin and point G is 1. Therefore, by using the distance formula, we have:
(sqrt(a^2 + b^2))^2 = 1
Simplifying further, we get:
a^2 + b^2 = 1
Now, we can use the fact that point G lies on the terminal side of angle theta in standard position. This means that the x-coordinate of G is equal to cos(theta), and the y-coordinate of G is equal to sin(theta).
Therefore, we can substitute a for cos(theta) and b for sin(theta) in the equation a^2 + b^2 = 1 to get:
cos^2(theta) + sin^2(theta) = 1
Knowing that sin^2(theta) + cos^2(theta) = 1 for any angle theta, we can conclude that sin(theta) = b.
Thus, the value of sin(theta) is b.