Triangle Scaling Worksheet Triangle Scaling Lab
Topic: Special Right Triangles to Unit Circle Ratios
Student Records
Name: __________________________
Date: ___________________________
The Ferris Wheel Challenge
Imagine a Ferris wheel with a radius of exactly 1 unit . To find your exact height above the center at different angles, we need to "shrink" our standard special right triangles so their longest side (the hypotenuse) is also 1.
1 Scaling the \( 45^\circ-45^\circ-90^\circ \)
Standard Ratios: Leg = \( x \), Hypotenuse = \( x\sqrt{2} \)
Goal: Scale the triangle so the Hypotenuse = 1.
1. Set \( x\sqrt{2} = 1 \)
2. Solve for \( x \):
Rationalize the denominator:
Leg Leg Hyp = 1
Scaled Leg Length: _________
2 Scaling the \( 30^\circ-60^\circ-90^\circ \)
Standard Ratios: Short Leg = \( x \), Long Leg = \( x\sqrt{3} \), Hypotenuse = \( 2x \)
Goal: Scale the triangle so the Hypotenuse = 1.
1. Set \( 2x = 1 \). Therefore, Short Leg (\( x \)) = ________
2. Calculate Long Leg (\( x\sqrt{3} \)):
Final Scaled Dimensions
Short Leg
\( \frac{1}{2} \)
Long Leg
\( \frac{\sqrt{3}}{2} \)
Reflect: Why is it important that the hypotenuse is exactly 1?
"The unit circle is just special triangles hiding in a circle."
Triangle Scaling Slides TRIANGLE SCALING
Building the bridge between right triangles and the unit circle.
Scaling
Unit Circle
The Ferris Wheel Challenge
Imagine you are on a Ferris Wheel with a radius of 1 unit.
As the wheel turns, your horizontal and vertical distance from the center changes.
Big Question:
"How can we use our knowledge of 30, 45, and 60 degree triangles to find our exact coordinates at any stop?"
1 Unit
Standard Ratios Refresher
\( 45^\circ - 45^\circ - 90^\circ \)
Leg: \( x \)
Leg: \( x \)
Hypotenuse: \( x\sqrt{2} \)
\( 30^\circ - 60^\circ - 90^\circ \)
Short Leg: \( x \)
Long Leg: \( x\sqrt{3} \)
Hypotenuse: \( 2x \)
"What happens when we force the hypotenuse to be exactly 1?"
1
Scaling the Isosceles Right Triangle
If \( x\sqrt{2} = 1 \)...
\( x = \frac{1}{\sqrt{2}} \)
Rationalize it!
\( x = \frac{\sqrt{2}}{2} \)
Every leg of a \( 45^\circ \) triangle with a hypotenuse of 1 is exactly \( \frac{\sqrt{2}}{2} \).
The "Unit" 45-45-90
\( \frac{\sqrt{2}}{2} \) \( \frac{\sqrt{2}}{2} \) 1
2
Scaling the 30-60-90 Triangle
If \( 2x = 1 \)...
Short Leg: \( x = \frac{1}{2} \)
Long Leg: \( x\sqrt{3} = \frac{\sqrt{3}}{2} \)
In a unit circle, the vertical side (opposite 30°) is \( 0.5 \) and the horizontal side is about \( 0.866 \).
The "Unit" 30-60-90
\( \frac{1}{2} \) \( \frac{\sqrt{3}}{2} \) 1 30°
Scaling Blueprint Summary
Angle Hypotenuse Horizontal Leg Vertical Leg \( 30^\circ \) 1 \( \frac{\sqrt{3}}{2} \) \( \frac{1}{2} \) \( 45^\circ \) 1 \( \frac{\sqrt{2}}{2} \) \( \frac{\sqrt{2}}{2} \) \( 60^\circ \) 1 \( \frac{1}{2} \) \( \frac{\sqrt{3}}{2} \)
These coordinates are the building blocks of the entire Unit Circle.
Radian Measure Reference Radian Measure Reference
Mathematical Standards & Tools
\( \theta = \frac{s}{r} \)
What is a Radian?
A radian is the measure of a central angle that intercepts an arc with a length equal to the radius of the circle.
The Universal Truth
Since a full circle has a circumference of \( 2\pi r \), there are exactly \( 2\pi \) radians in a full circle (\( 360^\circ \)).
Conversion Formulas
Degrees to Radians
\( \text{deg} \times \frac{\pi}{180} \)
Radians to Degrees
\( \text{rad} \times \frac{180}{\pi} \)
r
Arc length = r
1 rad
\( \approx 57.3^\circ \)
"A radian is measuring with the 'string' of the radius wrapped around the edge."
Common Conversions
Degrees Radians (Exact) Radians (Approx) Visual Location \( 30^\circ \) \( \frac{\pi}{6} \) 0.52 Small slice \( 45^\circ \) \( \frac{\pi}{4} \) 0.78 Perfect diagonal \( 60^\circ \) \( \frac{\pi}{3} \) 1.05 Steep slice \( 90^\circ \) \( \frac{\pi}{2} \) 1.57 Top Center \( 180^\circ \) \( \pi \) 3.14 Far Left \( 360^\circ \) \( 2\pi \) 6.28 Full Circle
Quick Check
Convert \( 120^\circ \) to radians:
Convert \( \frac{3\pi}{4} \) to degrees:
How many radians in \( 270^\circ \)?
First Quadrant Mapping Worksheet First Quadrant Blueprint
Constructing Coordinates for \( 0^\circ \leq \theta \leq 90^\circ \)
Unit Circle Mapping
Name: _________________
The Radius-1 Rule
On a circle with radius \( r = 1 \) , the standard trig definitions simplify beautifully. If we draw a right triangle inside the circle:
Cosine (\( x \)) \( \cos(\theta) = \frac{\text{adj}}{\text{hyp}} = \frac{x}{1} = x \)
Sine (\( y \)) \( \sin(\theta) = \frac{\text{opp}}{\text{hyp}} = \frac{y}{1} = y \)
Key takeaway:
Every coordinate on the unit circle is simply (cos, sin) .
y = sin x = cos
(x, y)
Calculating the Coordinates
Use your scaled triangle values from Lesson 1 to fill in the missing coordinates.
Angle (\( \theta \)) Radian \( x \) (Cosine) \( y \) (Sine) Coordinate \( (x, y) \) \( 0^\circ \) \( 0 \) ______ ______ \( (1, 0) \) \( 30^\circ \) \( \frac{\pi}{6} \) \( \frac{\sqrt{3}}{2} \) \( \frac{1}{2} \) ____________ \( 45^\circ \) \( \frac{\pi}{4} \) ______ ______ ____________ \( 60^\circ \) ______ ______ ______ ____________ \( 90^\circ \) \( \frac{\pi}{2} \) \( 0 \) \( 1 \) \( (0, 1) \)
What about Tangent?
Tangent is defined as \( \frac{\text{Opposite}}{\text{Adjacent}} \). Since Sine is Opposite and Cosine is Adjacent:
\( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} \)
Calculate \( \tan(30^\circ) \):
Calculate \( \tan(45^\circ) \):
Symmetry and Reflection Worksheet Symmetry & Reflection
Expanding the Unit Circle
II
I
III
IV
The ASTC Rule
The quadrant determines whether your trigonometric functions are positive or negative based on the signs of \( x \) and \( y \).
Quadrant II
(\(-x\), \(+y\))
Sine is \(+\)
Quadrant I
(\(+x\), \(+y\))
All are \(+\)
Quadrant III
(\(-x\), \(-y\))
Tangent is \(+\)
Quadrant IV
(\(+x\), \(-y\))
Cosine is \(+\)
A S T C
"All Students Take Calculus"
Finding the Reference Angle (\( \theta' \))
A reference angle is the smallest positive acute angle between the terminal side of an angle and the x-axis .
Quadrant II
\( 180^\circ - \theta \)
Quadrant III
\( \theta - 180^\circ \)
Quadrant IV
\( 360^\circ - \theta \)
Reflection Practice
The coordinate in Q1 for \( 60^\circ \) is \( (\frac{1}{2}, \frac{\sqrt{3}}{2}) \). Use reflection to find the others:
1. Reflection across Y-axis (Quadrant II)
Angle: \( 120^\circ \)
New Coordinate:
( ___, ___ )
2. Reflection through Origin (Quadrant III)
Angle: \( 240^\circ \)
New Coordinate:
( ___, ___ )
Unit Circle Mastery Worksheet Unit Circle Mastery
FINAL EVALUATION PHASE
Student Score
___ / 20
01. SCANNING COORDINATES
Identify the reference angle and the quadrant sign for each given angle.
\( 150^\circ \) Target Angle
Ref. Angle
Quadrant
\( \frac{4\pi}{3} \) Target Angle
Ref. Angle
Quadrant
02. EVALUATION LAB
Provide the exact value for each trigonometric function. No calculators permitted.
\( \cos(225^\circ) \)
\( \sin(300^\circ) \)
\( \tan(135^\circ) \)
\( \sin(\frac{5\pi}{6}) \)
\( \cos(\frac{7\pi}{4}) \)
\( \tan(\frac{4\pi}{3}) \)
03. CIRCLE BLUEPRINT
Complete the standard coordinates (x, y) for all key angles.
Unit Circle Blueprint Complete
Mathematics Geometry/Trigonometry Sequence
Unit Circle Teacher Guide Navigator's Master Key
Teacher Guide & Complete Unit Circle Reference
Lesson Series
Unit Circle Mastery
Instructional Strategy
The core of this unit is moving from memorization to construction . Students should understand that every value on the circle is a geometric reality, not just a number in a table.
Scaling First: Establish the "why" by showing that scaling the hypotenuse to 1 makes math easier.Radian Discovery: Avoid just giving the conversion formula. Show that \( \pi \) radians is halfway around because \( C = 2\pi r \).Reference Angles: Teach students to visualize the "bow-tie" shape for all reference angles.
Common Misconceptions
Confusing \( \frac{1}{2} \) and \( \frac{\sqrt{3}}{2} \) at \( 30^\circ/60^\circ \) angles. (Remind them that \( 0.5 < 0.866 \)).
Placing the reference angle against the y-axis instead of the x-axis.
Forgetting the negative sign in Quadrants II, III, and IV.
The Completed Blueprint
\( (0, 1) \)
\( (0, -1) \)
\( (1, 0) \)
\( (-1, 0) \)
\( 60^\circ: (\frac{1}{2}, \frac{\sqrt{3}}{2}) \)
\( 120^\circ: (-\frac{1}{2}, \frac{\sqrt{3}}{2}) \)
Pacing Guide
Days 1-2: Geometry/Scaling
Days 3-4: Unit Circle Const.
Day 5: Evaluation Speed
Lesson Phase Key Questions to Ask Checks for Understanding Scaling Lab "If the hypotenuse is 10, how does that relate to a hypotenuse of 1?" Rationalizing \( \frac{1}{\sqrt{2}} \) correctly. Radian Revolution "Why do we use \( \pi \) in the measurement? What is \( \pi \)'s relation to a circle?" Conversion drill: \( 90^\circ, 180^\circ, 270^\circ \). Quadrant Mapping "If sine is the y-value, why is it positive in Quadrant II?" Finding \( \tan(45^\circ) \) by dividing \( y/x \). Circle Mastery "How can you find \( \cos(330^\circ) \) using only Quadrant I?" Rapid-fire evaluation of mixed rad/deg angles.
