LEARNER JOURNEY Example 1: sin30°
PointP30=(√3/2,1/2).
SelectSine reads y, the second coordinate.
Resultsin30°=1/2.
Check30° is above the x-axis, so sine is positive.
Start here
Build the concept from the beginning
Starting point
No prior trigonometry is assumed. Every required term and decision is taught on this page.
Learning target
Understand sine as vertical position on the unit circle, including its sign and range.
Evidence of mastery
Construct the reasoning, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- sine
- The y-coordinate of the unit-circle terminal point.
- vertical projection
- The up-down component from the x-axis to the point.
- range
- All possible sine values, from −1 through 1.
- zero of sine
- An angle whose terminal point lies on the x-axis.
Look first, then name the mathematics
Build the visual meaning
- sin θ = y
- Vertical coordinate
- radius = 1
- P = (x, y)
Read the complete visual relationship as text
- Sine reads the second coordinate.
- Above-axis points have positive sine.
- Below-axis points have negative sine.
- Left and right axis points have sine 0.
Definition in plain language
Follow the terminal side to P=(x,y). Read the second coordinate: sin θ=y. The radius is 1, so −1≤sin θ≤1.
Why this matters
This definition explains signs, exact values, and why sine tracks vertical rather than horizontal movement.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Find the terminal point or its quadrant/reference angle.
Construct or read the ordered pair (x,y).
Select the second coordinate y.
Check its sign from above/below and its magnitude against [−1,1].
The same method in words
- Find the terminal point or its quadrant/reference angle.
- Construct or read the ordered pair (x,y).
- Select the second coordinate y.
- Check its sign from above/below and its magnitude against [−1,1].
Interactive decision model
Build the reasoning, one decision at a time
Construct sin 315° from the unit-circle point.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Each example names the decision, carries it out, and interprets the result.
Example 2: sin240°
Reference240° has reference 60°.
PointP=(−1/2,−√3/2).
SelectRead y.
Resultsin240°=−√3/2, negative below the x-axis.
Example 3: sin180°
Locate180° ends at (−1,0).
SelectThe y-coordinate is 0.
Resultsin180°=0.
InterpretThe left point has no vertical displacement.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. Which coordinate defines sine?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. What is the range of sine?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Complete: if P=(−3/5,4/5), then sinθ=____.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Independent practice
Six problems for repetition
Solve all six without opening the worked examples unless feedback shows what to revise.
1. Find sin0°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Find sin90°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Find sin270°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Find sin45°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Find sin120°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
6. Find sin330°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. What is the sign of sine in Quadrant II, and why?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Find sin225°.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Find sin(3π/2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. If sinθ=0, where is the terminal point?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Why can sinθ never equal −1.4 on the unit circle?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Evaluate sin(5π/3) from the definition, explaining the angle, point, selected coordinate, and sign check.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
5π/3=300°, which lies in Quadrant IV with reference angle 60°. The exact point is (1/2,−√3/2). Sine is the y-coordinate, so sin(5π/3)=−√3/2. The negative sign is correct because the terminal point is below the x-axis.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.