Circle Master Teacher Guide

Lesson: From SOHCAHTOA to x,y | Grade 10-12 Mathematics

Duration 50 MIN

Objective

Students will transition from right-triangle trigonometry to coordinate geometry by defining \(\sin(\theta)\) as the \(y\)-coordinate and \(\cos(\theta)\) as the \(x\)-coordinate on a unit circle. They will analyze how scaling the radius affects coordinates but preserves trigonometric ratios.

Materials Needed

Mapping the Circle Worksheet
Trig Transition Slides
Circle Scaling Grid (Graph Paper)
Scientific Calculators
Trig Talk Cards

Key Vocabulary

Unit Circle

Radius = 1, centered at (0,0).

Standard Position

Angle starting at positive x-axis, opening CCW.

Reciprocal

Multiplicative inverse (used for tangent calculation).

Lesson Flow

00-05 min

Warm-up: The \(30^\circ\) Shift

Students draw a standard right triangle for \(30^\circ\) on their worksheet. They must identify \(\sin(30^\circ)\) using SOHCAHTOA. Check for understanding: Ensure they label the hypotenuse as 1 to prime them for the unit circle.

05-20 min

Video Synthesis

Play the video: Unit Circle Definition of Trigonometric Functions. Pause at the following points:

[5:27] Pause for students to calculate/fill in \(45^\circ\) and \(60^\circ\) coordinates on their worksheet.
[8:17] "Your Turn" pause. Have students vote on the angle before the answer is revealed.

20-30 min

The "Why" Discussion

Facilitate using the Trig Talk Cards. Focus on: "Why is radius 1 better than radius 2?" (Simplifies the denominator in SOHCAHTOA to 1).

30-45 min

Scaling Up Activity

Using the Circle Scaling Grid, students plot points on a circle with radius 5. They compare these to the unit circle.
Goal: See that while the \(x\) and \(y\) coordinates grow by a factor of 5, the ratio \(\frac{y}{r}\) remains constant (\(\sin(\theta)\)).

45-50 min

3-2-1 Reflection

Students complete the reflection on their worksheet: 3 things learned, 2 questions, 1 key takeaway.

Answer Key

Unit Circle Coordinates

\(30^\circ\): \((\frac{\sqrt{3}}{2}, \frac{1}{2})\)
\(45^\circ\): \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\)
\(60^\circ\): \((\frac{1}{2}, \frac{\sqrt{3}}{2})\)

Scaling Up (Radius 5)

Coordinates at \(30^\circ\): \((5\cos 30^\circ, 5\sin 30^\circ) = (\frac{5\sqrt{3}}{2}, 2.5)\)
Verification: \(\frac{y}{r} = \frac{2.5}{5} = 0.5\). Same as \(\sin(30^\circ)\).

Common Misconception

Students often swap Sine and Cosine for \(x\) and \(y\). Remedy: Reinforce the alphabetical mnemonic: Cosine (X) comes before Sine (Y).

SOHCAHTOA to XY

Circle Master Teacher Guide

Mapping the Circle Worksheet

Trig Transition Slides

Circle Scaling Grid

Trig Talk Cards

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