Question Video: Exploring the Different Angles between 0 and 2𝜋 That Have the Same Trigonometric Function | Nagwa Question Video: Exploring the Different Angles between 0 and 2𝜋 That Have the Same Trigonometric Function | Nagwa

# Question Video: Exploring the Different Angles between 0 and 2π That Have the Same Trigonometric Function Mathematics • First Year of Secondary School

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Suppose π is a point on a unit circle corresponding to the angle of 4π/3. Is there another point on the unit circle representing an angle in the interval [0, 2π) that has the same tangent value? If yes, give the angle.

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### Video Transcript

Suppose π is a point on the unit circle corresponding to the angle four π over three. Is there another point on the unit circle representing an angle in the interval zero to two π that has the same tangent value? If yes, give the angle.

First, we might wanna sketch a coordinate plane and then add a unit circle, which is a circle with the center at the origin and a radius of one. From there, we might also want to label our coordinate plane with radians beginning at zero, π over two, π, three π over two, and two π. Because we have the interval zero to two π, we know weβre only interested in one full turn. Our point π is on the unit circle and corresponds to the angle four π over three. This means our first job is to find out where the angle four π over three would land.

I know for π over three is greater than π, but itβs probably worth comparing to find out if four π over three is greater than or less than three π over two. If we give these fractions common denominators, four π over two becomes eight π over six and three π over two becomes nine π over six. Since eight π over six is less than nine π over six, we can say that four π over three is less than three π over two. And that means point π is going to fall in our third quadrant and that four π over three would be this angle.

Since we know that our angle falls in the third quadrant, we can use the CAST diagram, which will tell us that the tangent of the angle in the third quadrant is going to be positive. In order for us to find another point inside the unit circle that has the same tangent value, weβll be looking for the other place where the tangent value could be positive. And that will be in the first quadrant. In the first quadrant, all trig values will be positive.

But in order for us to find what the value of that angle would be in the first quadrant, we need to break up our angle four π over three into smaller parts. We could say that four π over three is equal to π plus π over three, the distance from zero to π and then an additional π over three. The right triangle created inside the unit circle four π over three in the third quadrant would look like this.

And in the first quadrant, there would be some point such that we would be dealing with the angle of π over three. In the first quadrant, that would have π₯, π¦ coordinates. And in the third quadrant, it would have negative π₯, negative π¦ coordinates. And we know that in a unit circle, the tan of π will be equal to π¦ over π₯. And we would say that the tan of four-thirds π is equal to negative π¦ over negative π₯ and the tan of π over three is equal to π¦ over π₯. But we simplify negative π¦ over negative π₯ to just π¦ π₯. And so, weβve shown that yes, there is another angle in this interval that has the same tangent value. And itβs the angle π over three.

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