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Learning Resource Type

Lesson Plan

Unit Circle - Special Angles - Just Know One

Subject Area



9, 10, 11, 12


This lesson will demonstrate that in order to find the coordinates of the special angles on the unit circle, students will need a knowledge of the first quadrant angles only. Students will use special right triangle relationships for 30° - 60° -90° or 45° - 45° - 90° triangles to find the first quadrant coordinate values. These values will then be reflected across the x- and y-axis to locate the coordinates in the remaining quadrants. Students will also convert the angle measurements from units in degrees to units in radians.  They will become familiar with finding angles in the quadrants by using reference angles (π-x, π+x. 2π-x).

This lesson results from the ALEX Resource Gap Project.


    Use special triangles to determine geometrically the values of sine, cosine, and tangent for $frac {pi}{3}$, $frac {pi}{4}$, and $frac {pi}{6}$, and use the unit circle to express the values of sine, cosine, and tangent for $pi - x$, $pi + x$, and $2pi - x$ in terms of their values for $x$, where $x$ is any real number.

    Primary Learning Objectives

    Students will identify the relationship of the sides of a 30° - 60° -90° and a 45° - 45° - 90° triangle.

    Students will find the coordinates of the angles on the unit circle in quadrant I by using special right triangle relationships.

    Students will convert angles in degree measurement to radian measurement.

    Students will apply their knowledge of reflecting objects in the coordinate plane across the x-axis and the y-axis to find the coordinates of special angles in quadrants II, III and IV.

    Students will find reference angles for special angles in quadrants II, III, IV. They will use π-x, π+x. 2π-x.



    Pass out the Special Right Triangles Review worksheet. This is intended to remind students of the relationship between the sides of special right triangles 30° - 60° -90° and 45° - 45° - 90°. They are asked to use the Pythagorean Theorem to find a missing side and then to look at the result to remember the relationship. (This is a skill from Geometry, therefore, this activity is intended as simply a review of the topic.)

    Allow students time to talk with other students if they are not seeing or recalling the relationship.

    When students have completed the worksheet review student findings with the whole group. Explain the relationships to be certain all students have identified them correctly.

    Be sure to write or post the relationships on the board as these will be used in the lesson.

    30° - 60° -90°: 1:√3:2 (longest side is 2 times the length of the shortest side and the middle length [across from the 60°] is √3 times the shortest side)

    45° - 45° - 90°: 1:1:√2 (the hypotenuse is √2 times the equal legs [isosceles triangle])


    1.  Open the PowerPoint-Unit Circle Quadrant Values. Students will find and record answers to the questions on each slide. First, ask students to find a radian measure that is equal to the degree measure of 30°.

    2.  Next, have the students use the relationship of the sides of special right triangles to find the lengths of the sides of the 30° - 60° - 90° triangle. Remind them that they are working with the Unit Circle, so we know that the radius is 1 unit. They should be able to find the horizontal and vertical sides of the triangle which will become the x- and y-values for the coordinate point. You will want to point out that the horizontal side will correspond to the Cosine value and the vertical side will correspond to the Sine value.

    3. Repeat steps 1-2 for the second slide 45°.

    4. Finally repeat steps 1-2 for the third slide 60°.

    5.  Next show students slide 4. This slide contains the points that they found in the first three slides. Explain that you only need to know the coordinates of the points in the first quadrant. Have students take notes as you show them that the remaining quadrants can be thought of as a reflection of points across either the x- or y-axis (depending on the quadrant). If necessary, draw attention to the fact that the point is always either 30°, 45° or 60° from the x-axis. Therefore, the sides will be the same as the sides of the triangles in the first quadrant. Also, point out that the angles' measurements can be found by either subtracting x from π, adding x to π or subtracting x from 2π.

    6.  Pair students together and pass out "I Have, Who Has?" cards. It may be necessary for some pairs to have more than one card. Instruct students to take their "I have" statement and be sure that they are able to identify the degree measure, the radian measure, and the coordinate points. If they have an angle in quadrants II, III or IV, they should also recognize the angle expressed as π-x, π+x or 2π-x.  

    7.  After students have all needed information ask the pair of students who has the "Start Here" card to go first. They should be able to self check as the first group reading their "I Have" statement should also be the last group to answer the "Who Has?" question.


    Ask students to sketch and fill in the first quadrant of the unit circle. (A free printable blank copy can be found here.) Next, have them fill in the other quadrants using the information from quadrant I.

    Assessment Strategies


    Teachers will check student work as they circulate the classroom during the Special Right Triangle review.  

    Working in pairs will allow the students to assist each other in finding degree measures, radian measures and coordinate points for the special angles on the unit circle. Teachers will post answers to allow students to assess themselves.

    Students will assess themselves as they participate in the I Have, Who Has card game.  


    Teachers will collect the unit circle as an exit ticket and check for student understanding.


    Advanced students can also be given non-special angles and be asked to find Reference angles using (π - x, π + x, or 2π - x).  Practice can be found here.


    Struggling students may need extra review to recall the special right triangle relationships.  Extra practice can be found here.

    Extra practice on converting degree measures to radian measures can be found here.

    Approximate Duration

    Total Duration

    31 to 60 Minutes

    Background and Preparation



    Students will be asked to draw on knowledge of special right triangles (30° - 60° -90° and 45° - 45° - 90°). They will also be asked to convert degree measures to radian measures.  

    Students will need to know how to participate in an I Have, Who Has card game.


    Teachers will need to know how to use PowerPoint and be familiar with the Unit Circle - Quadrant Values presentation.  

    Teachers will need to be familiar with an I Have, Who Has card game.

    Make copies of the Special Right Triangle Review (one copy per student or projected on board for students to view).


    Materials and Resources

    Materials and Resources



    Technology Resources Needed

    Computer connected to projector

    PowerPoint - Unit Circle Quadrant Values

    Computer with internet connection for Optional Assignment for Advanced or struggling students (Special Right Triangle Review or Converting Radians Review)