Noesis
3.1.1Advanced Trigonometry

Unit circle definition of trig functions — all 6 trig functions for any angle

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🇮🇳 Yeh note Hinglish mein padho →

WHAT: the setup

The whole trick: that landing point defines everything.

WHY does this match right-triangle trig? Drop a vertical line from PP to the x-axis. You get a right triangle with hypotenuse = radius = 11, horizontal leg =x=x, vertical leg =y=y. Then cos⁡θ=adjacenthyp=x1=x,sin⁡θ=oppositehyp=y1=y.\cos\theta = \frac{\text{adjacent}}{\text{hyp}} = \frac{x}{1} = x, \qquad \sin\theta = \frac{\text{opposite}}{\text{hyp}} = \frac{y}{1} = y. So for 0°<θ<90°0°<\theta<90° the old SOH-CAH-TOA answer is exactly the coordinate. The circle just keeps going where the triangle stops.

HOW: the other four functions (derived, not memorized)

Everything else is built from sin⁡\sin and cos⁡\cos. Derive from ratios:

WHY these definitions? In the right triangle, tan⁡=oppadj=yx\tan = \frac{\text{opp}}{\text{adj}} = \frac{y}{x}. Secant/cosecant/cotangent are just the reciprocals — "co-" partners with "co-", so sec⁡\sec pairs with cos⁡\cos (both start the ratio from the x-side) — actually memorize by reciprocal, see mnemonic.

Key consequence — where they blow up:

  • tan⁡θ, sec⁡θ\tan\theta,\ \sec\theta are undefined when x=cos⁡θ=0x=\cos\theta=0 (i.e. θ=90°,270°,…\theta = 90°, 270°, \dots).
  • cot⁡θ, csc⁡θ\cot\theta,\ \csc\theta are undefined when y=sin⁡θ=0y=\sin\theta=0 (i.e. θ=0°,180°,…\theta = 0°, 180°, \dots).

You can see this: division by zero when a coordinate hits the axis.

Figure — Unit circle definition of trig functions — all 6 trig functions for any angle

The Pythagorean identity — for free

Signs by quadrant

As you rotate, xx and yy change sign. So the trig functions do too.

Quadrant angle range x=cos⁡x=\cos y=sin⁡y=\sin positive functions
I 0°0°–90°90° ++ ++ All
II 90°90°–180°180° −- ++ Sine (+csc)
III 180°180°–270°270° −- −- Tan (+cot)
IV 270°270°–360°360° ++ −- Cos (+sec)

Negative and huge angles

Worked examples

Common mistakes

Recall Feynman: explain to a 12-year-old

Imagine a merry-go-round with radius exactly 11 meter, and you start at the "3 o'clock" spot. When you spin to some angle, look at where you are: how far right you are (that's cos⁡\cos) and how far up you are (that's sin⁡\sin). If you go past the top, "how far right" becomes negative because you're now on the left side. Spin all the way around and you're back where you started — that's why the numbers repeat. The other four functions (tan⁡\tan, etc.) are just these two divided by each other. When you're exactly at the top or side and one of them is zero, dividing by it "breaks the calculator" — that's why some are undefined.

Flashcards

What is cos⁡θ\cos\theta on the unit circle?
The x-coordinate of the point you reach after rotating θ\theta from (1,0)(1,0).
What is sin⁡θ\sin\theta on the unit circle?
The y-coordinate of that point.
Define tan⁡θ\tan\theta from coordinates.
tan⁡θ=y/x=sin⁡θ/cos⁡θ\tan\theta = y/x = \sin\theta/\cos\theta.
Why is sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1?
Because (x,y)(x,y) lies on the circle x2+y2=1x^2+y^2=1, and x=cos⁡θx=\cos\theta, y=sin⁡θy=\sin\theta.
When is tan⁡θ\tan\theta undefined and why?
When cos⁡θ=x=0\cos\theta=x=0 (θ=90°,270°,…\theta=90°,270°,\dots) — division by zero.
When is csc⁡θ\csc\theta undefined?
When sin⁡θ=y=0\sin\theta=y=0 (θ=0°,180°,…\theta=0°,180°,\dots).
In which quadrants is sin⁡θ\sin\theta positive?
Q1 and Q2 (upper half, y>0y>0).
In which quadrant are only tan⁡\tan and cot⁡\cot positive?
Q3 (both x,y<0x,y<0).
Is cosine even or odd?
Even: cos⁡(−θ)=cos⁡θ\cos(-\theta)=\cos\theta (same x when rotating clockwise).
Is sine even or odd?
Odd: sin⁡(−θ)=−sin⁡θ\sin(-\theta)=-\sin\theta (y-coordinate mirrors).
Value of sin⁡90°\sin90° and cos⁡90°\cos90°?
sin⁡90°=1\sin90°=1, cos⁡90°=0\cos90°=0 (point (0,1)(0,1)).
How do you handle sin⁡(390°)\sin(390°)?
Subtract 360°360° (periodicity): sin⁡390°=sin⁡30°=1/2\sin390°=\sin30°=1/2.
Mnemonic for positive functions per quadrant?
"All Students Take Calculus": All, Sine, Tan, Cos.
sec⁡θ=?\sec\theta = ? in coordinates.
1/x=1/cos⁡θ1/x = 1/\cos\theta.

Connections

  • Right-triangle trigonometry (SOH-CAH-TOA) — the special case 0°<θ<90°0°<\theta<90°.
  • Reference angles — how to reduce any angle to an acute one.
  • Pythagorean identities — all three come from x2+y2=1x^2+y^2=1.
  • Radian measure — angles as arc length on the unit circle.
  • Graphs of trig functions — sine/cosine graphs are the y/x coordinates plotted vs θ\theta.
  • Even and odd functions — cosine even, sine odd.
  • Periodicity and $2\pi$ — full-loop invariance.

Concept Map

start on

lands at

x-coordinate

y-coordinate

ratio sin/cos

ratio sin/cos

reciprocal

reciprocal

division by zero

division by zero

renamed identity

substitute into circle

substitute into circle

matches for 0-90 deg

matches for 0-90 deg

x,y change sign

Unit circle x2+y2=1

Rotate by theta from 1,0

Landing point P = x,y

cos theta = x

sin theta = y

tan = y/x

sec, csc, cot reciprocals

Undefined when coord = 0

sin2 + cos2 = 1

Right-triangle SOH-CAH-TOA

Signs by quadrant

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, right-angle triangle se hum sirf 0°0° se 90°90° tak ke angles handle kar sakte hain. Lekin hume kabhi sin⁡(150°)\sin(150°) ya cos⁡(270°)\cos(270°) bhi chahiye hota hai. Iske liye aata hai unit circle — ek circle jiska radius bilkul 11 hai, center origin par. Aap (1,0)(1,0) point se start karo aur θ\theta angle jitna ghumo (counterclockwise = positive). Jahan aap pahunchte ho, us point ke coordinates hi trig functions ban jaate hain: x-coordinate = cos⁡θ\cos\theta, y-coordinate = sin⁡θ\sin\theta. Bas itni si baat hai.

Baaki chaar functions inhi do se bante hain: tan⁡=y/x\tan = y/x, cot⁡=x/y\cot = x/y, sec⁡=1/x\sec = 1/x, csc⁡=1/y\csc = 1/y. Jab bhi xx ya yy zero hota hai (jaise top point (0,1)(0,1) par x=0x=0), tab divide-by-zero ke kaaran function undefined ho jaata hai. Aur ek badhiya baat — kyunki point circle par hi hai, x2+y2=1x^2+y^2=1 automatically ban jaata hai, jiska matlab sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1. Yeh identity yaad karne ki zaroorat nahi, yeh to circle ka equation hi hai naya naam le kar.

Signs ke liye "All Students Take Calculus" yaad rakho — Q1 mein sab positive, Q2 mein sirf sin, Q3 mein tan, Q4 mein cos. Negative angle ka matlab clockwise ghumna, aur 360°360° add karna matlab poora ek chakkar laga kar wahi point — isliye functions repeat hote hain (periodic). Isse har real angle ke liye trig define ho jaata hai, chahe wo −1000°-1000° hi kyun na ho.

Go deeper — visual, from zero

D1FoundationsEvery symbol from zero · 2,046 wordsD2Visual walkthroughThe derivation in pictures · 2,285 wordsD3Worked examplesEvery scenario covered · 3,686 wordsD4ExercisesGraded, full solutions · 2,103 wordsD5Question bankConcept traps, quick reveals · 1,832 words

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Connections

Reference anglesMaths · 3.1.5Pythagorean identities — sin² + cos² = 1, derivations of the other twoMaths · 3.1.9Radian measure — definition, conversion formula degrees ↔ radiansMaths · 3.1.2Graphs of functions — plotting, reading key featuresMaths · 2.2.6Even and odd functions — graphical and algebraic testsMaths · 2.2.10Trigonometric ratios in right triangle — sin, cos, tan, cosec, sec, cotMaths · 2.4.2SOH-CAH-TOA mnemonicMaths · 2.4.3Transformations of trig graphs — A·sin(Bx + C) + D (amplitude, period, phase, vertical shift)Maths · 3.1.8Sum and difference formulas — sin(A±B), cos(A±B), tan(A±B) — proofsMaths · 3.1.12