Unit circle and trigonometric functions of any angle

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MODULE: Trigonometric Functions and Identities

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✖️ 1. Defining the unit circle and terminal point coordinates

🎯 The Unit Circle Blueprint

  • The unit circle is a circle with radius 1 centered at the origin.
  • Its equation is x2+y2=1x^2 + y^2 = 1.
  • Any angle θ\theta measured from the positive x-axis creates a terminal point (x,y)(x, y) on the circle.
  • The coordinates are always (x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos \theta, \sin \theta).
  • The x-coordinate gives cosine, the y-coordinate gives sine.

Example: At θ=0\theta = 0 degrees, the terminal point is (1,0)(1, 0), so cos⁡0=1\cos 0 = 1 and sin⁡0=0\sin 0 = 0.

💡 Think: x = cos, y = sin — the circle remembers the trig values for you!

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1. Defining the unit circle and terminal point coordinates

Defining the Unit Circle and Terminal Point Coordinates

The unit circle is the set of all points (x,y)(x, y) in the coordinate plane satisfying x2+y2=1x^2 + y^2 = 1, centered at the origin with radius 1. For any angle θ\theta measured counterclockwise from the positive xx-axis, the terminal point where the angle's terminal side intersects the unit circle has coordinates (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta).

This definition extends trigonometric functions beyond right triangles to any real angle.

Core Rules:

  • The xx-coordinate of the terminal point equals cos⁡θ\cos \theta
  • The yy-coordinate of the terminal point equals sin⁡θ\sin \theta
  • All points satisfy cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1 (Pythagorean identity)
  • Positive angles rotate counterclockwise; negative angles rotate clockwise

This coordinate representation allows trigonometric evaluation for angles of any magnitude, not restricted to acute angles.

Example: For θ=90∘\theta = 90^\circ, the terminal point is (0,1)(0, 1), so cos⁡90∘=0\cos 90^\circ = 0 and sin⁡90∘=1\sin 90^\circ = 1.

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MOD: TRANSLATE

The terminal point of an angle θ\theta on the unit circle is (0.6,0.8)(0.6, 0.8). What is the value of cos⁡θ\cos \theta?

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✖️ 2. Symmetry and periodicity of the unit circle for full rotations

🔄 Circle Repeats Forever

  • A full rotation is 360 degrees or 2π2\pi radians.
  • After one full turn, the terminal point returns to the same position.
  • This means sin⁡(θ+360∘)=sin⁡θ\sin(\theta + 360^\circ) = \sin \theta and cos⁡(θ+360∘)=cos⁡θ\cos(\theta + 360^\circ) = \cos \theta.
  • Trig functions repeat every 360 degrees (they are periodic).
  • The circle has symmetry across both axes and the origin.

Example: sin⁡30∘=sin⁡390∘=sin⁡750∘\sin 30^\circ = \sin 390^\circ = \sin 750^\circ because 390=30+360390 = 30 + 360 and 750=30+2(360)750 = 30 + 2(360).

💡 Visual: Spin the circle — same angle, same point, same values!

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2. Symmetry and periodicity of the unit circle for full rotations

Symmetry and Periodicity of the Unit Circle

Periodicity means that adding full rotations (multiples of 360∘360^\circ or 2π2\pi radians) to any angle produces the same terminal point. The unit circle exhibits rotational symmetry, causing trigonometric functions to repeat their values at regular intervals.

This property arises because a complete rotation returns to the starting position on the circle.

Core Rules:

  • sin⁡(θ+360∘)=sin⁡θ\sin(\theta + 360^\circ) = \sin \theta and cos⁡(θ+360∘)=cos⁡θ\cos(\theta + 360^\circ) = \cos \theta (period is 360∘360^\circ or 2π2\pi)
  • sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta (odd function, symmetry about origin)
  • cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \theta (even function, symmetry about yy-axis)
  • Reflections across axes relate function values in different quadrants

Periodicity enables simplification of angles outside [0∘,360∘)[0^\circ, 360^\circ) by subtracting multiples of 360∘360^\circ.

Example: sin⁡450∘=sin⁡(450∘−360∘)=sin⁡90∘=1\sin 450^\circ = \sin(450^\circ - 360^\circ) = \sin 90^\circ = 1.

TASK_1[0 / 3]
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Find the exact value of cos⁡(720∘)\cos(720^\circ).

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✖️ 3. Understanding reference angles and extending trigonometric definitions to all four quadrants

🧭 Reference Angles Unlock All Quadrants

  • A reference angle is the acute angle between the terminal side and the x-axis.
  • It is always between 0 and 90 degrees.
  • Use the reference angle to find trig values in any quadrant.
  • Sign rules: Quadrant I (both +), II (sin +, cos −), III (both −), IV (sin −, cos +).
  • Find the reference angle, then apply the correct sign based on the quadrant.

Example: For θ=150∘\theta = 150^\circ (Quadrant II), reference angle is 30∘30^\circ. So sin⁡150∘=+sin⁡30∘=12\sin 150^\circ = +\sin 30^\circ = \frac{1}{2} and cos⁡150∘=−cos⁡30∘=−32\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}.

💡 Memory: "All Students Take Calculus" — which functions are positive in each quadrant!

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3. Understanding reference angles and extending trigonometric definitions to all four quadrants

Reference Angles and Extension to All Quadrants

A reference angle is the acute angle formed between the terminal side of a given angle and the nearest xx-axis. It allows computation of trigonometric values in any quadrant by relating them to familiar acute angle values, adjusted for sign based on quadrant.

Reference angles preserve absolute values while quadrant determines sign.

Core Rules:

  • Reference angle θr\theta_r is always between 0∘0^\circ and 90∘90^\circ
  • Quadrant I: both sine and cosine positive
  • Quadrant II: sine positive, cosine negative
  • Quadrant III: both negative
  • Quadrant IV: sine negative, cosine positive
  • ∣sin⁡θ∣=sin⁡θr|\sin \theta| = \sin \theta_r and ∣cos⁡θ∣=cos⁡θr|\cos \theta| = \cos \theta_r

This method systematically extends trigonometric evaluation beyond the first quadrant.

Example: For θ=150∘\theta = 150^\circ (Quadrant II), reference angle is 30∘30^\circ, so sin⁡150∘=sin⁡30∘=1/2\sin 150^\circ = \sin 30^\circ = 1/2 and cos⁡150∘=−cos⁡30∘=−3/2\cos 150^\circ = -\cos 30^\circ = -\sqrt{3}/2.

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What is the reference angle in degrees for an angle of 210∘210^\circ?

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✖️ 4. Evaluating exact trigonometric values for standard angles

📐 Standard Angles You Must Know

  • Memorize exact values for 30, 45, 60 degrees and their multiples.
  • At 30 degrees: sin⁡30∘=12\sin 30^\circ = \frac{1}{2}, cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}.
  • At 45 degrees: sin⁡45∘=cos⁡45∘=22\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}.
  • At 60 degrees: sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}, cos⁡60∘=12\cos 60^\circ = \frac{1}{2}.
  • Use reference angles and quadrant rules to find values like 120, 135, 150, 210, 225, 240, 300, 315, 330 degrees.

Example: cos⁡240∘\cos 240^\circ is in Quadrant III (reference angle 60 degrees), so cos⁡240∘=−cos⁡60∘=−12\cos 240^\circ = -\cos 60^\circ = -\frac{1}{2}.

💡 Pattern: 30-60-90 triangle gives 12,32\frac{1}{2}, \frac{\sqrt{3}}{2}; 45-45-90 gives 22\frac{\sqrt{2}}{2}!

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4. Evaluating exact trigonometric values for standard angles

Evaluating Exact Trigonometric Values for Standard Angles

Standard angles are specific angles (30∘30^\circ, 45∘45^\circ, 60∘60^\circ, and their multiples) whose trigonometric values can be expressed exactly using radicals. These values derive from special right triangles: the 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle and the 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle.

Memorizing these values enables rapid exact computation without calculators.

Core Rules:

  • sin⁡30∘=1/2\sin 30^\circ = 1/2, cos⁡30∘=3/2\cos 30^\circ = \sqrt{3}/2
  • sin⁡45∘=cos⁡45∘=2/2\sin 45^\circ = \cos 45^\circ = \sqrt{2}/2
  • sin⁡60∘=3/2\sin 60^\circ = \sqrt{3}/2, cos⁡60∘=1/2\cos 60^\circ = 1/2
  • For multiples, use reference angles and quadrant sign rules
  • Values at 0∘0^\circ, 90∘90^\circ, 180∘180^\circ, 270∘270^\circ are 00, ±1\pm 1

These exact values form the foundation for analytical trigonometry.

Example: cos⁡240∘\cos 240^\circ has reference angle 60∘60^\circ in Quadrant III, so cos⁡240∘=−cos⁡60∘=−1/2\cos 240^\circ = -\cos 60^\circ = -1/2.

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Evaluate the exact value of sin⁡150∘\sin 150^\circ.

Enter your answer as a decimal.

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✖️ 5. Applications in simple harmonic motion and AC voltage cycles

⚡ Real-World Waves and Cycles

  • Simple harmonic motion (like a swinging pendulum) follows y=Asin⁡(ωt)y = A \sin(\omega t) or y=Acos⁡(ωt)y = A \cos(\omega t).
  • AC voltage alternates in a sine wave pattern: V(t)=V0sin⁡(2πft)V(t) = V_0 \sin(2\pi f t).
  • The unit circle models periodic behavior — anything that repeats in cycles.
  • Amplitude AA is the maximum displacement; frequency ff controls how fast the cycle repeats.
  • One complete cycle corresponds to one full rotation around the unit circle.

Example: If AC voltage is V(t)=120sin⁡(120πt)V(t) = 120 \sin(120\pi t) volts, the peak voltage is 120 volts and frequency is 60 Hz (since 2πf=120π2\pi f = 120\pi).

💡 Think: Sine waves = circular motion viewed from the side!

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5. Applications in simple harmonic motion and AC voltage cycles

Applications: Simple Harmonic Motion and AC Voltage Cycles

Simple harmonic motion and alternating current (AC) voltage are modeled using sinusoidal functions derived from circular motion on the unit circle. As a point rotates uniformly around the circle, its xx and yy coordinates oscillate sinusoidally, representing physical quantities varying periodically over time.

The unit circle provides the geometric foundation for these periodic phenomena.

Core Rules:

  • Position in harmonic motion: x(t)=Acos⁡(ωt+ϕ)x(t) = A \cos(\omega t + \phi) where AA is amplitude, ω\omega is angular frequency, ϕ\phi is phase
  • AC voltage: V(t)=V0sin⁡(ωt)V(t) = V_0 \sin(\omega t) where V0V_0 is peak voltage
  • Period T=2π/ωT = 2\pi/\omega represents one complete cycle
  • Frequency f=1/Tf = 1/T measures cycles per unit time

These models predict oscillatory behavior in mechanical systems and electrical circuits.

Example: If AC voltage is V(t)=120sin⁡(120πt)V(t) = 120\sin(120\pi t) volts, the peak voltage is 120 volts and frequency is 6060 Hz.

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RSN: PATTERN

An alternating current circuit has a voltage modeled by the function V(t)=170sin⁡(120πt)V(t) = 170 \sin(120\pi t).

What is the peak voltage of this circuit in volts?

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