# Question Video: Explaining the Extension of Trigonometric Functions to All Real Numbers with the Unit Circle Mathematics

Consider a windmill with the blades of length 1 m. The position of the top 𝑃 of a given blade is given by the coordinates (𝑎, 𝑏) which depends on the angle 𝜃. Express 𝑎 and 𝑏 as functions of the measure of angle 𝜃 in radians. If angle 𝜃 at a certain time is 5𝜋/3, what will it be after the blade has completed half a rotation?

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### Video Transcript

Consider a windmill with the blades of length one meter. The position of the top 𝑃 of a given blade is given by the coordinates 𝑎, 𝑏, which depends upon the angle 𝜃 as shown. Express 𝑎 and 𝑏 as functions of the measure of angle 𝜃 in radians. If angle 𝜃 at a certain time is five-thirds 𝜋, what will it be after the blade has completed half a rotation?

Because the length of the blades are one meter and in our diagram, this represents the radius of the windmill, we can use our knowledge of the unit circle to help us solve this problem. The distance from the center of the windmill to point 𝑃 would be one. And we’re told that the point 𝑃 is located at 𝑎, 𝑏. And so, we can create a right triangle with the 𝑥-axis and say that its side lengths are 𝑎 and 𝑏, respectively. We use the distance from the terminal side to the 𝑥-axis to calculate the angle 𝜃. We know that in the unit circle, we can represent the angle as a sine and cosine relationship. For our angle 𝜃, the opposite side length would be 𝑏 and its hypotenuse is one meter, so we have sin of 𝜃 equals 𝑏 over one. And we can represent 𝑏 as sin of 𝜃.

Similarly, if we look at cosine, we end up with the side length 𝑎 over one, which means we can say that the distance 𝑎 must be equal to the cos of the angle 𝜃. And without more information, this is as far as we can go with these two functions. We can say that 𝑏 equals sin of 𝜃 and 𝑎 equals cos of 𝜃.

Part two of our questions says if the angle of 𝜃 at a certain time is five 𝜋 over three, what will it be after the blade has completed a half a rotation? First of all, we’ve already been told that we’re operating in radians, and so it might be helpful to label our coordinate plane. Beginning at the 𝑥-axis, zero radians, then 𝜋 over two radians, 𝜋 radians, three 𝜋 over two radians, and a full turn, which is two 𝜋 radians. In a system like this, a full turn is two 𝜋. And therefore, a half turn is going to be equal to 𝜋 radians.

If we start with the angle 𝜃 five 𝜋 over three radians and we add a half a rotation, we’re adding 𝜋 to that angle. And just like adding any fractions, we need a common denominator. We can write 𝜋 as three 𝜋 over three. We add five 𝜋 plus three 𝜋 to get eight 𝜋. And the denominator wouldn’t change. The angle 𝜃 after a half rotation would be eight 𝜋 over three.

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