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Defining Sine, Cosine, and Tangent for Any Angle (Lesson 6.3)

Unit 6 Day 3
CED Topic(s):

​Learning Targets​
  • Extend the definition of sine, cosine, and tangent ratios to angles greater than 90˚ using the coordinate plane and horizontal and vertical displacement.

  • Understand why in a unit circle, the sine and cosine ratios correspond with the y-value and x-value, respectively, of the point where the terminal ray intersects the circle.

  • Understand that in a unit circle, the tangent of an angle is the ratio of the y-coordinate to the x-coordinate of the point where the terminal ray intersects the circle. Alternately, the tangent ratio is the slope of the terminal ray.

  • Use symmetry to identify relationships between the sine, cosine, and tangent values of angles in all four quadrants.

Quick Lesson Plan
Activity: Trig Ratios in the Wild



Lesson Handout

Answer Key



Experience First:

For this lesson, students will need rulers and protractors (yes, we’re breaking the no degree rule–but it’s for a good cause!). This is the first lesson of the unit where students are actually using sine, cosine, and tangent. We expect students to come in with an understanding of right triangle trigonometry but we will build the concept of trig values for angles greater than π/2 from the ground up. 


On the first page of the Activity, students are given a diagram of three concentric circles on the coordinate grid and asked to draw a 40˚ angle. (We break the radian only rule in this lesson because we need students to look at all kinds of angles and be able to measure them with a protractor.)

The terminal ray of this angle intersects all three circles. What makes this lesson unique is its focus on distances, or in the case of other quadrants, displacement from the x and y axis, rather than just coordinates. Students are very familiar with the idea that a trig ratio represents a ratio of side lengths in a right triangle. When defining trig ratios for angles that could not possibly be in a right triangle (angles greater than 90˚) we want to preserve the idea that a trig value represents a ratio of segments. The grid paper gives students the opportunity to estimate horizontal and vertical distances, rather than using a calculator or only being able to look at angles that come from special right triangles. 


In question 4 and 5 students encounter another essential idea of trigonometry: similar triangles. The triangles created by points A, B, and C all have the same angles so they are similar triangles. Because they are similar triangles, all triangles will produce the same ratios of sides. Because the ratios of specific sides are equal in all three triangles, any of the three triangles can be used to determine sin(40˚) and cos (40˚). Is any triangle favorable? Hmmm…


Page 2 now has students look at an angle in the second quadrant. Now they find the horizontal and vertical displacement which takes into account the direction from the x and y axis, not just the distance. Using the idea of displacement gives meaning to the sine, cosine, and tangent values in quadrants 2-4, rather than just telling students to “adjust for the quadrant.”


In question 10 students are asked to articulate why, of all the similar right triangles, the triangle with a hypotenuse of 1 is most helpful (i.e. distances measured from a circle of radius 1).


In question 11, be looking for different methods students are using to describe and determine the tangent value. Are they comparing vertical distance to horizontal distance? Are they dividing the y coordinate by the x coordinate? Are they dividing the sine value by the cosine value? Are they finding the slope made by the terminal ray? These are all equivalent conceptions of the tangent, and the more of them you can bring up in the debrief, the better!

Monitoring Questions:
  • How did you find the distance from the origin?

  • What do you notice about the right triangles made by A, B, and C?

  • How do we define sin(150˚) if there isn’t a right triangle we can draw with an angle of 150˚?

  • Why did we use distance and not displacement on the first page?

  • In which quadrants are the vertical displacements negative?

  • In which quadrants are the horizontal displacements negative?

  • Is the distance from the origin ever negative?

  • In which quadrants will the tangent value be negative? Why?

  • Where is this new point that has the same horizontal displacement but opposite vertical displacement?

  • What can you say about the sine, cosine, and tangent of 210˚?

Formalize Later:

This lesson will touch on right triangle trigonometry concepts (SOHCAHTOA) when dealing with angles in the first quadrant, but if you find students don’t have a good conceptual understanding of what sine, cosine, and tangent represent IN RIGHT TRIANGLES, we suggest doing the Precalculus 4.1 Math Medic lesson before returning to this lesson.


I have always struggled with the quick turn around between right triangle trigonometry and the “sin=y, cos= x” mantra students recite over and over once they learn about the unit circle. It seems like students abandon the rich conceptual learning we did around ratios in similar right triangles once they are asked to evaluate sine, cosine, and tangent values. For most students the unit circle becomes a memorization game. This lesson seeks to provide the middle (and necessary) step between lengths of sides in a right triangle and coordinates on a circle. That middle step is horizontal and vertical displacements. Ratio of lengths, rather than ratios of coordinates. The debrief will then make the connection that x and y coordinates of a point are the horizontal and vertical displacements of the point from both axes. The debrief should also emphasize why the unit circle is helpful, but not necessary, for evaluating trig ratios. Any sized circle will do–but mathematicians often look for efficient strategies, and what’s more efficient than a denominator that is always 1?


Before teaching this lesson, make sure you are very familiar with the intent of this lesson. We are using a somewhat unique sequence in this course in order to establish better conceptual understanding, and it can be easy to slip into our old familiar patterns. Here are a few key things to keep in mind.

  • This lesson is NOT about finding exact values for sine, cosine and tangent.

  • This lesson is NOT about “key points” on the unit circle. 

  • This lesson is about defining sine, cosine, and tangent ratios for angles greater than 90˚ (a wild idea if you live in right triangle land).

  • This lesson is about using circles on the coordinate plane to extend the domain of trigonometric functions to all real numbers.

  • This lesson is about understanding why we use the unit circle, rather than any other circle, to evaluate trig functions.

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