Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Find a reference angle.
- Assign signs using the quadrant.
- Scale unit-circle coordinates by radius.
Before you start
The 30°–60°–90° triangle, square roots, and coordinate quadrants.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A radius-4 marker is at a standard-position angle of 150°. Find its coordinates and the sine, cosine, and tangent of its angle.
Why this math matters
Right-triangle ratios begin with acute angles. The unit circle extends sine and cosine to complete rotations and attaches their signs to horizontal and vertical position.
Set up the model
A useful answer starts with clear assumptions:
- The centre is the origin and the initial ray is the positive x-axis.
- Positive rotation is counterclockwise.
- The point lies exactly 4 units from the origin.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Use the unit circle beyond acute angles
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A radius-4 marker is at a standard-position angle of 150°. Find its coordinates and the sine, cosine, and tangent of its angle.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Find the reference angle
180° − 150° = 30°
The terminal ray lies in quadrant II, thirty degrees above the negative x-axis.
Assign exact values
cos150° = −√3/2; sin150° = 1/2
A 30° reference supplies the magnitudes. Quadrant II gives negative horizontal position and positive vertical position.
Scale and form tangent
(x, y) = 4(cos150°, sin150°) = (−2√3, 2); tan150° = −1/√3
Radius multiplies both coordinates. Tangent is sine divided by cosine, which is defined because cosine is nonzero.
The result
The marker is at (−2√3, 2); sine is 1/2, cosine is −√3/2, and tangent is −√3/3.
The coordinates satisfy x² + y² = 12 + 4 = 16, confirming the radius. A negative coordinate indicates direction, not a negative radial distance.
Common mistakes to catch
- Reference-angle values still need quadrant signs.
- Unit-circle coordinates must be multiplied by 4 for this circle.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find the unit-circle coordinates at 225°.
Show a hint
Use reference angle 45° in quadrant III.
Reveal answer and explanation
(−√2/2, −√2/2)
Both coordinates are negative and have the 45° magnitude.
Practice 2
Solve sin θ = 1/2 for 0° ≤ θ < 360°.
Show a hint
Sine is positive above the x-axis.
Reveal answer and explanation
30° and 150°
Both first- and second-quadrant rays have vertical coordinate 1/2.
Take the idea with you
Use coordinates to reason about signs instead of memorizing signs independently from the diagram.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Estimate a tree's height with trigonometry
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