Maths with Lemon

Using the Unit Circle

IB Mathematics: Analysis and Approaches · SL 3.4 and AHL 3.4. Explore exact trigonometric values, reference angles, and the connections between sine, cosine and tangent.

Download Lesson 5.2 worksheet (PDF) ↗

The Unit Circle

What you need to know

  • Coordinates in the Cartesian plane
  • Angles in degrees and radians
  • Special right triangles: 30°–60°–90° and 45°–45°–90°

Start with the AA Geometry & Trigonometry topic overview.

Key Points

The unit circle is centred at the origin and has radius 1. The point where the terminal side of an angle meets the circle has coordinates (cos θ, sin θ).

\[x^2+y^2=1\qquad (x,y)=(\cos\theta,\sin\theta)\]
\[\tan\theta=\frac{\sin\theta}{\cos\theta}\quad (\cos\theta\ne0)\]
Angles are measured counterclockwise when positive and clockwise when negative.

Explore Angles and Coordinates

Practice ideas

  • Predict the signs of sine and cosine before moving the slider
  • Compare angles differing by 360°
  • Find the reference angle for a negative angle
  • Explain when tangent is undefined

Use the interactive circle alongside your Lesson 5.2 worksheet.

Interactive Unit Circle

Move the slider to see how the coordinates change. The horizontal coordinate is cosine; the vertical coordinate is sine.

xy 1−11 (cos θ, sin θ)
cos θ0.707
sin θ0.707
tan θ1.000

Try 135°, −120°, 390° and 690° from your worksheet.

The slider shows decimal approximations. In IB exam questions, use exact values whenever requested.

Exact Values and Special Angles

What you need to know

  • Convert degrees to radians
  • Recall exact sine, cosine and tangent values
  • Identify 0°, 30°, 45°, 60° and 90° on the circle

Key Points

These are the essential first-quadrant values. Other angles use the same reference angles, with signs determined by the quadrant.

Angle0°30°45°60°90°
Radians\(0\)\(\pi/6\)\(\pi/4\)\(\pi/3\)\(\pi/2\)
sin θ\(0\)\(1/2\)\(\sqrt2/2\)\(\sqrt3/2\)\(1\)
cos θ\(1\)\(\sqrt3/2\)\(\sqrt2/2\)\(1/2\)\(0\)
tan θ\(0\)\(\sqrt3/3\)\(1\)\(\sqrt3\)Undefined
Worked example: 210°

The reference angle is 30°. Since 210° is in quadrant III, sine and cosine are negative while tangent is positive.

\[\sin210^\circ=-\frac12,\quad\cos210^\circ=-\frac{\sqrt3}{2},\quad\tan210^\circ=\frac{\sqrt3}{3}\]

Quadrants, Reference Angles and Coterminal Angles

What you need to know

  • Signs of trigonometric ratios in all four quadrants
  • Reference angles
  • Angles outside 0° to 360°
  • Negative angles and coterminal angles

Quadrant Signs

Quadrant I

sin +, cos +, tan +

Quadrant II

sin +, cos −, tan −

Quadrant III

sin −, cos −, tan +

Quadrant IV

sin −, cos +, tan −

Worked example: 390°

Subtract one complete revolution to find a coterminal angle.

\[390^\circ-360^\circ=30^\circ\]
\[\sin390^\circ=\frac12,\quad\cos390^\circ=\frac{\sqrt3}{2},\quad\tan390^\circ=\frac{\sqrt3}{3}\]
Worked example: −7π/6

Add \(2\pi\) to find a positive coterminal angle:

\[-\frac{7\pi}{6}+2\pi=\frac{5\pi}{6}\]

The reference angle is \(\pi/6\), in quadrant II.

\[\sin\theta=\frac12,\quad\cos\theta=-\frac{\sqrt3}{2},\quad\tan\theta=-\frac{\sqrt3}{3}\]

Trigonometric Identities from the Circle

Skills

  • Relate angles in different quadrants
  • Use periodicity
  • Explain why cosine is even and sine is odd
  • Recognise tangent as a ratio

Key Points

Reflections and rotations of the unit circle explain many identities:

\[\cos(-x)=\cos x\qquad \sin(-x)=-\sin x\]
\[\sin(\pi+x)=-\sin x\]
\[\tan(3\pi-x)=-\tan x\]
\[\sin(x+2\pi)=\sin x\qquad\cos(x+2\pi)=\cos x\]
Your Lesson 5.2 syllabus table specifically highlights relationships between angles in different quadrants.

HL Extension: Reciprocal Trigonometric Functions

HL only

  • Cosecant, secant and cotangent
  • Undefined values and domain restrictions
  • Exact reciprocal values

These exercises appear on page 3 of your uploaded worksheet.

Key Points

\[\csc\theta=\frac1{\sin\theta},\quad\sec\theta=\frac1{\cos\theta},\quad\cot\theta=\frac1{\tan\theta}\]
Worked example: θ = π/3

Start with the exact primary ratios:

\[\sin\frac\pi3=\frac{\sqrt3}{2},\quad\cos\frac\pi3=\frac12,\quad\tan\frac\pi3=\sqrt3\]

Take reciprocals and rationalise where useful:

\[\csc\frac\pi3=\frac{2\sqrt3}{3},\quad\sec\frac\pi3=2,\quad\cot\frac\pi3=\frac{\sqrt3}{3}\]
Remember: \(\csc heta\) is undefined when \(\sin heta=0\); \(\sec heta\) is undefined when \(\cos heta=0\); and \(\cot heta\) is undefined when \(\sin heta=0\).

Extra: Practice and References

Materials

For the PDF link to work, upload the worksheet to the same directory as this HTML page, or update its href to the actual server path.

Revision Checklist

Coordinates

Identify \((\cos heta,\sin heta)\).

Exact values

Use special angles without a calculator.

Reference angles

Handle negative and large angles.

HL

Calculate cosecant, secant and cotangent.

Worksheet practice: complete the warm-up circle on page 1, six angle questions on page 2, and the HL reciprocal ratios on page 3.

IB Past Paper Problems

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