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Coordinates on the Unit Circle (Lesson 6.4)

Unit 6 Day 4
CED Topic(s):

​Learning Targets​
  • Use special right triangles to determine the coordinates at key points on the unit circle.

  • Evaluate sine, cosine, and tangent for key angles on the unit circle.

  • Find coordinates of points on circles where r≠1.

Quick Lesson Plan
Activity: Coming Full Circle



Lesson Handout

Answer Key



Experience First:

In the previous lesson students learned that the sine, cosine, and tangent ratios can be determined by the coordinates of the point on the unit circle where the terminal ray of the angle intersects the circle. Students estimated the values of these coordinates. How do we find exact coordinates? This is the focus of today’s lesson.


In questions 1 and 2, the students will write ratios for sine and cosine when given a special right triangle with all of the side lengths. We want them to see that when the hypotenuse is 1, the actual side length of the triangles are the same as the sine and cosine ratios for specific angles. This will lead them to the unit circle, where the radius is 1 and the values of sine and cosine at different angles are equal to the x- and y-coordinates at those points. The idea here is that students don’t get caught up in “Sine = y and cosine = x,” but they keep up with the understanding that sine is the ratio of the opposite side to the hypotenuse and cosine is the ratio of the adjacent side to the hypotenuse. This idea was introduced yesterday as students considered different sized circles on the coordinate plane and reinforced today as students consider special right triangles of various sizes. 


On page 2, students use reflected special right triangles to find the ordered pairs at key points on the unit circle and then use these ordered pairs to evaluate the sine, cosine, and tangent functions. The big take-away is that for angles that are multiples of π/6 or π/4, the x and y coordinates of the terminal ray point do not have to be estimated, they can be found exactly!

Monitoring Questions:
  • What’s the same about these three (isosceles) triangles? What’s different?

  • How many times greater is the hypotenuse than the leg in this 45˚, 45˚, 90˚ triangle? 

  • What is special about the triangle where the hypotenuse is 1?

  • What is the same about the ordered pairs at these 4 points? What is the same about the angles made by the terminal ray of these 4 points?

  • Yesterday you estimated sin (150˚) and cos(150˚). Now can you find their exact values?

  • Why is it important to be able to estimate the x and y coordinate?

  • Why is it important to have an exact value for the x and y coordinate? 

Formalize Later:

There are not a lot of margin notes in this lesson so the debrief will be fairly short. You may wish to have students work only on page 1 and then debrief question 1 before having them move to the back page.


While there are many memorization tricks out there for the coordinates of key unit circle points, we encourage students to build and use flexible strategies rather than relying on tricks. If students memorize one thing it would be the lengths of the sides in the two special right triangles where the hypotenuse is 1. Understanding how these can be placed on the unit circle to find the coordinates is then sufficient for determining the sine, cosine, and tangent of any angle. If students are confusing the x and y coordinates at​​ π/6 or π/3, help them decide whether the x-coordinate or the y-coordinate at that angle is greater.  This applies to any point on the unit circle, though comparing distance from the x-axis and distance from the y-axis is more accurate for angles in quadrants 2-4. It can also be helpful for students to see that the coordinates in each quadrant increase or decrease in a predictable way. Writing the values as sqrt(0)/2, then sqrt(1)/2, then sqrt(2)/2, then sqrt(3)/2, and finally sqrt(4)/2 helps students clearly see how the values compare to each other.

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