Thirty-Six

Unit 7 · TrigonometryG6 · Trigonometry · Aerodynamics & Navigation

Unit Circle, Degrees & Radians

Read exact unit-circle values, track signs by quadrant, and convert angle units.

  • 28 min
  • Unlocks: Star Tracker
  • Desmos way

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

unit circle
A radius-1 circle whose point at angle theta is (cos theta, sin theta).
radian
An angle unit based on arc length; pi radians equals 180 degrees.
reference angle
The acute angle to the x-axis.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

Reduce the angle to a familiar reference angle. Then use the quadrant sign pattern. Desmos starts in degrees on ACT, so switch to radians only when the input contains radians.

Key idea

The decision that matters

On the unit circle, x is cosine and y is sine. The reference triangle gives the size; the quadrant gives the sign.

Degree-radian bridge
θdeg180∘=θradπ\frac{\theta_{\mathrm{deg}}}{180^\circ}=\frac{\theta_{\mathrm{rad}}}{\pi}

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

At 150∘150^\circ, the reference angle is 30∘30^\circ. Quadrant II makes cosine negative and sine positive, so the point is (−32,12)\left(-\frac{\sqrt3}{2},\frac12\right).

Explore in Desmos

Connect angles to unit-circle coordinates

Keep degree mode on. Click each plotted point and read its decimal coordinates. Change both angles to 330° and compare the signs of cosine (x) and sine (y) with the quadrant.

The calculator is live: change anything and see what happens.

Connect angles to unit-circle coordinates

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

Loading Desmos…
CueBest move
Need coordinatesx is cosine; y is sine
Need signsUse the quadrant after the reference angle

Trap

Using the right value with the wrong sign

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1: Reference angle and signs

Problem

Find the unit-circle point for 210∘210^\circ.
The reference angle is 210∘−180∘=30∘210^\circ-180^\circ=30^\circ.

The 30∘30^\circ magnitudes are 3/2\sqrt3/2 for cosine and 1/21/2 for sine.

Which coordinates have the correct reference values and signs?

The angle is in Quadrant III, where both coordinates are negative.

The point is (−3/2,−1/2)(-\sqrt3/2,-1/2).

Why are both coordinates negative at 210 degrees?

Worked example

Example 2: Convert radians to degrees

Problem

Convert 7π/67\pi/6 radians to degrees.
Multiply by the unit fraction 180∘/π180^\circ/\pi.

7π6⋅180∘π=7(30∘)=210∘\frac{7\pi}{6}\cdot\frac{180^\circ}{\pi}=7(30^\circ)=210^\circ.

Which conversion factor changes radians to degrees?

The π\pi units cancel; the result is in degrees.

Why does the conversion factor cancel radians?

Faded example · you finish it

Example 3: You finish it

Problem

Find cos⁡(330∘)\cos(330^\circ). The reference angle is 30∘30^\circ.
Find the reference angle, then attach the signs required by the quadrant.

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.