Question Video: Finding the Value of a Trigonometric Function given the Coordinates of the Point of Intersection of the Unit Circle with the Terminal Side of an Angle in Standard Position Mathematics

Video Transcript

The terminal side of angle 𝐴𝑂𝐵 in standard position intersects with the unit circle at the point 𝐵 with coordinates negative 𝑥, negative 𝑥, where 𝑥 is a positive number. Find sin of 𝜃.

First, let’s sketch a unit circle to help us visualize what’s happening here. The unit circle has a center at the origin and a radius of one. Our coordinate 𝐵 is negative 𝑥, negative 𝑥, where 𝑥 is a positive number. This means that our point 𝐵 will be in the third quadrant, 𝑥 units to the left of the origin and 𝑥 units below the origin. Since angle 𝐴𝑂𝐵 is in standard position, its initial side is the positive 𝑥-axis and its terminal side will go through the point 𝐵.

And for angles in standard position, they’ll be calculated counterclockwise from the origin. Before we can calculate the sin of 𝜃, we’ll calculate what’s called a reference angle from point 𝐵. This is the angle created with point 𝐵 and the negative 𝑥-axis. Let’s call this angle 𝛼. From there, we can use our knowledge of right triangle trigonometry to find the angle 𝛼. This right triangle has two side lengths that both measure 𝑥 and a hypotenuse of one. We know that the hypotenuse is one because point 𝐵 falls on the unit circle, and the distance between 𝐵 and the center of this circle is one.

To solve for 𝑥, we can use the Pythagorean theorem, which says 𝑎 squared plus 𝑏 squared equals 𝑐 squared. In our triangle, this is 𝑥 squared plus 𝑥 squared equals one squared. Two 𝑥 squared equals one. Dividing both sides by two, we find that 𝑥 squared equals one over two. Taking the square root of both sides, we find that 𝑥 equals the square root of one over two, which simplifies to one over the square root of two. We know that the sin ratio is equal to the opposite side length over the hypotenuse. And therefore, the sin of 𝛼 will be equal to 𝑥 over one, or simply 𝑥. This makes the sin of 𝛼 one over the square root of two.

And this is the sin for our reference angle. In order to find accurately the sin of our 𝜃 relationship with the initial side being the 𝑥-axis and the terminal side being point 𝐵, we use a CAST diagram. In the first quadrant, all trig values are positive. In the second quadrant, only the sine value is positive. In the third quadrant, only the tangent value is positive. And in the fourth quadrant, the cosine value is positive.

Since our point 𝐵, our terminal side, is in quadrant three, the sin of angle 𝐴𝑂𝐵 will be negative. We use the absolute value of our reference angle, and the CAST diagram tells us that sin of 𝜃 is negative. The sin of 𝜃 equals negative one over the square root of two.