Circular Functions of an Angle – Module 2 Study Notes

Learning Objectives

  • Define and become familiar with the six primary circular (trigonometric) functions.
  • Determine the algebraic sign ((+) or (-)) of each function in every quadrant.
  • Transform (reduce) a trigonometric expression of any angle into one that involves its acute reference angle.
  • Apply knowledge of special angles ((30^\circ,\;45^\circ,\;60^\circ)) and quadrantal angles to find exact values of the six functions.
  • Sketch the graphs of sine, cosine, tangent, cosecant, secant, and cotangent.

The Six Circular (Trigonometric) Functions

  • Sine (()\sin\theta=\dfrac{y}{r})
  • Cosine (()\cos\theta=\dfrac{x}{r})
  • Tangent (()\tan\theta=\dfrac{y}{x})
  • Cosecant (()\csc\theta=\dfrac{r}{y}=\dfrac{1}{\sin\theta})
  • Secant (()\sec\theta=\dfrac{r}{x}=\dfrac{1}{\cos\theta})
  • Cotangent (()\cot\theta=\dfrac{x}{y}=\dfrac{1}{\tan\theta})

Signs of the Functions in the Four Quadrants (ASTC Rule)

  • Quadrant I ((0^\circ<\theta<90^\circ)) : All six functions are positive.
  • Quadrant II ((90^\circ<\theta<180^\circ)) : Sine and cosecant are positive; others negative.
  • Quadrant III ((180^\circ<\theta<270^\circ)) : Tangent and cotangent are positive; others negative.
  • Quadrant IV ((270^\circ<\theta<360^\circ)) : Cosine and secant are positive; others negative.

(Mnemonic: "All Students Take Calculus")

Reduction to a Function of an Acute Angle

  • Any angle (\theta) can be rewritten in terms of its reference angle (\alpha) ((0^\circ<\alpha<90^\circ)).
  • General rule:
    (sin⁡θ,cos⁡θ,tan⁡θ,csc⁡θ,sec⁡θ,cot⁡θ)=±(sin⁡α,cos⁡α,tan⁡α,csc⁡α,sec⁡α,cot⁡α)(\sin\theta,\cos\theta,\tan\theta,\csc\theta,\sec\theta,\cot\theta)=\pm(\sin\alpha,\cos\alpha,\tan\alpha,\csc\alpha,\sec\alpha,\cot\alpha)
  • The (\pm) sign is chosen using the ASTC rule.
How to Find the Reference Angle
  1. Locate the terminal side of (\theta) on the coordinate plane.
  2. Drop a perpendicular to the x–axis; the acute angle between this line and the x–axis is (\alpha).
  3. Use quadrant logic:
    • QI: (\alpha = \theta)
    • QII: (\alpha = 180^\circ-\theta)
    • QIII: (\alpha = \theta-180^\circ)
    • QIV: (\alpha = 360^\circ-\theta)
      (Replace (180^\circ) and (360^\circ) with (\pi) and (2\pi) in radian measure.)
Worked Reductions

  • Example 1 (Given): Express the six functions of (420^\circ).

    • Find coterminal angle: (420^\circ-360^\circ=60^\circ) ⇒ Quadrant I ⇒ sign (+).
    • Reference angle (\alpha=60^\circ).
    • sin⁡420∘=sin⁡60∘,  cos⁡420∘=cos⁡60∘,  tan⁡420∘=tan⁡60∘,  csc⁡420∘=csc⁡60∘,  sec⁡420∘=sec⁡60∘,  cot⁡420∘=cot⁡60∘.\sin420^\circ=\sin60^\circ,\;\cos420^\circ=\cos60^\circ,\;\tan420^\circ=\tan60^\circ,\;\csc420^\circ=\csc60^\circ,\;\sec420^\circ=\sec60^\circ,\;\cot420^\circ=\cot60^\circ.

  • Example 2 (Given): Express the six functions of (-135^\circ).

    • Coterminal: (-135^\circ+360^\circ=225^\circ) (Quadrant III).
    • Reference angle: (225^\circ-180^\circ=45^\circ).
    • Sign pattern in QIII ⇒ sine (-), cosine (-), tangent (+).
    • sin⁡(−135∘)=−sin⁡45∘,cos⁡(−135∘)=−cos⁡45∘,\sin(-135^\circ)=-\sin45^\circ,\quad \cos(-135^\circ)=-\cos45^\circ,
      tan⁡(−135∘)=+tan⁡45∘,\tan(-135^\circ)=+\tan45^\circ,
      csc⁡(−135∘)=−csc⁡45∘,  sec⁡(−135∘)=−sec⁡45∘,  cot⁡(−135∘)=+cot⁡45∘.\csc(-135^\circ)=-\csc45^\circ,\;\sec(-135^\circ)=-\sec45^\circ,\;\cot(-135^\circ)=+\cot45^\circ.
Practice Items (from text)
  • Exercise 1: "Express (\tan\left(-\dfrac{\pi}{4}\right)) as a function of a positive acute angle."
  • Exercise 2: "Reduce (\sec(-240^\circ15')) to the same function of a positive acute angle." (Procedure identical—find coterminal angle, quadrant, sign.)

Quadrantal Angles

  • Definition: Angles whose terminal side lies exactly on the x– or y–axis.
  • Standard counter-clockwise set: (0^\circ,\;90^\circ,\;180^\circ,\;270^\circ,\;360^\circ).
  • Clockwise set: (0^\circ,\;-90^\circ,\;-180^\circ,\;-270^\circ,\;-360^\circ).
  • Coordinates on the unit circle:
    • (0^\circ\,(1,0))
    • (90^\circ\,(0,1))
    • (180^\circ\,(-1,0))
    • (270^\circ\,(0,-1))
  • Resulting exact values:
    • sin⁡0∘=0,  cos⁡0∘=1,  tan⁡0∘=0,\sin0^\circ=0,\;\cos0^\circ=1,\;\tan0^\circ=0,
    • sin⁡90∘=1,  cos⁡90∘=0,  tan⁡90∘=undefined,\sin90^\circ=1,\;\cos90^\circ=0,\;\tan90^\circ=\text{undefined},
    • sin⁡180∘=0,  cos⁡180∘=−1,  tan⁡180∘=0,\sin180^\circ=0,\;\cos180^\circ=-1,\;\tan180^\circ=0,
    • sin⁡270∘=−1,  cos⁡270∘=0,  tan⁡270∘=undefined,\sin270^\circ=-1,\;\cos270^\circ=0,\;\tan270^\circ=\text{undefined},
    • Similar values for other functions can be derived (cosecant, secant undefined when sine or cosine equals 0).
Suggested Table to Complete (Exercise 3)
  • Create a 6 × 4 table listing (\sin,\cos,\tan,\csc,\sec,\cot) versus the four quadrant angles above.
Numerical Evaluation (Exercise 4)

a) & b) (Actual expressions not displayed in transcript. Solve by substituting tabulated values.)

Special Angles and Their Exact Function Values

FamilyReference‐Angle DiagramExact Values (Unit Circle)
(30^\circ\;\bigl(\dfrac{\pi}{6}\bigr))30-60-90 trianglesin⁡=12,  cos⁡=32,  tan⁡=13\sin=\dfrac{1}{2},\;\cos=\dfrac{\sqrt{3}}{2},\;\tan=\dfrac{1}{\sqrt{3}} csc⁡=2,  sec⁡=23,  cot⁡=3\csc=2,\;\sec=\dfrac{2}{\sqrt{3}},\;\cot=\sqrt{3}
(45^\circ\;\bigl(\dfrac{\pi}{4}\bigr))Isosceles right trianglesin⁡=cos⁡=22,  tan⁡=1\sin=\cos=\dfrac{\sqrt{2}}{2},\;\tan=1 csc⁡=sec⁡=2,  cot⁡=1\csc=\sec=\sqrt{2},\;\cot=1
(60^\circ\;\bigl(\dfrac{\pi}{3}\bigr))same 30-60-90 trianglesin⁡=32,  cos⁡=12,  tan⁡=3\sin=\dfrac{\sqrt{3}}{2},\;\cos=\dfrac{1}{2},\;\tan=\sqrt{3} csc⁡=23,  sec⁡=2,  cot⁡=13\csc=\dfrac{2}{\sqrt{3}},\;\sec=2,\;\cot=\dfrac{1}{\sqrt{3}}

(Memorize these; all other exact values come from them via signs & reference angles.)

Visual Memory Aids
  • "Hand Trick" for sine/cosine values (folding fingers to count (0^\circ,30^\circ,45^\circ,60^\circ,90^\circ)).
  • Unit-circle poster summarizing all 16 special + quadrantal angles.

Fundamental Pythagorean Identity

  • sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1 (true (\forall\theta))
Example 3 (Given)
  • Evaluate sin⁡2θ+cos⁡2θ\sin^2\theta+\cos^2\theta when (\theta=45^\circ).
    • (\sin45^\circ=\cos45^\circ=\dfrac{\sqrt{2}}{2}).
    • (22)2+(22)2=12+12=1.\Bigl(\dfrac{\sqrt{2}}{2}\Bigr)^2+\Bigl(\dfrac{\sqrt{2}}{2}\Bigr)^2=\dfrac{1}{2}+\dfrac{1}{2}=1.
    • Confirms identity.

Graphing Sketch Guidelines

  1. Sine: starts at ((0,0)), amplitude 1, period 2π2\pi.
  2. Cosine: starts at ((0,1)), amplitude 1, same period.
  3. Tangent: vertical asymptotes at (\dfrac{\pi}{2}+k\pi), period π\pi.
  4. Cosecant/Secant: Reciprocal graphs of sine/cosine; same asymptotes where originals cross x–axis.
  5. Cotangent: Vertical asymptotes at (k\pi); period π\pi.
Practical Sketch Steps
  • Mark the x–axis in multiples of (\pi/2) or (90^\circ).
  • Draw baseline (midline) for sine/cosine.
  • Plot key points derived from special angles.
  • Extend using periodicity.

Ethical & Practical Considerations

  • Materials are for educational use only; redistribution without permission is prohibited.
  • Mastery of exact values saves calculator dependence, vital in fields where estimation speed matters (engineering, physics teaching).

Real‐World Connections

  • Circular functions model oscillations (sound waves), rotational motion (gears), periodic phenomena (seasons, tides).
  • Knowledge of signs per quadrant is crucial for vector resolution and phasor diagrams in AC electricity.

Summary Checklist for Self-Study

  • [ ] Memorize the ASTC quadrant sign mnemonic.
  • [ ] Master exact values for (30^\circ,45^\circ,60^\circ,90^\circ,0^\circ).
  • [ ] Practice reducing arbitrary angles (degrees & radians) to acute reference angles.
  • [ ] Complete the quadrantal-angle table without notes.
  • [ ] Redraw the six primary function graphs from memory.
  • [ ] Verify understanding through textbook Exercises 1–4.