Complementary angle relationships — sin(90−θ) = cos θ etc.

2.4.5 · Maths › Trigonometry — Foundation

Overview

Complementary angles do angles hote hain jinका sum 90° hota hai. Complementary angles ke trigonometric ratios mein ek khoobsurat symmetry hoti hai: sine aur cosine swap ho jaate hain, tangent aur cotangent swap ho jaate hain, secant aur cosecant swap ho jaate hain. Inhe cofunction identities kehte hain kyunki har pair mein ek function aur uska "co-" partner hota hai.

The Six Cofunction Identities

$\sin (90°-\theta )=\cos \theta$ $\cos (90°-\theta )=\sin \theta$ $\tan (90°-\theta )=\cot \theta$ $\cot (90°-\theta )=\tan \theta$ $\sec (90°-\theta )=\csc \theta$ $\csc (90°-\theta )=\sec \theta$

Radians mein: 90° ki jagah π/2 likho.

Derivation from First Principles

From Right Triangle Definitions

Setup: Ek right triangle consider karo jisme acute angles θ aur (90° − θ) hain.

Step 1: Triangle ko label karo.

• Maano hypotenuse = h
• θ ke opposite side = a (to yeh angle 90° − θ ke adjacent side hai)
• θ ke adjacent side = b (to yeh angle 90° − θ ke opposite side hai)

Yeh kyun important hai: Wahi side alag-alag roles play karti hai depending on karte ho ki tum kis angle se measure kar rahe ho.

Step 2: Definition use karke sin(90° − θ) likho.

$\sin (90°-\theta )=\frac{\text{opposite to }(90°-\theta )}{\text{hypotenuse}}=\frac{b}{h}$

Yeh step kyun? Hum definition apply karte hain: sine = opposite/hypotenuse for angle (90° − θ).

Step 3: Pehchano ki b/h cos θ hai.

$\cos \theta =\frac{\text{adjacent to }\theta }{\text{hypotenuse}}=\frac{b}{h}$

Yeh step kyun? Side b angle θ ke adjacent hai, isliye b/h exactly cos θ ki definition hai.

Step 4: Identity conclude karo.

$\sin (90°-\theta )=\frac{b}{h}=\cos \theta$

Yeh kyun kaam karta hai: Geometry swap ko force karti hai. Ek angle ke perspective se opposite side complementary angle ke perspective se adjacent side hoti hai.

Deriving the Other Identities

Usi triangle se: $\cos (90°-\theta )=\frac{\text{adjacent to }(90°-\theta )}{h}=\frac{a}{h}=\sin \theta$

(90° − θ) ke adjacent side wahi side hai jo θ ke opposite hai.

Sine aur cosine results use karke: $\tan (90°-\theta )=\frac{\sin (90°-\theta )}{\cos (90°-\theta )}=\frac{\cos \theta }{\sin \theta }=\cot \theta$

Yeh step kyun? Tangent define hota hai sine/cosine se. Hum woh identities substitute karte hain jo humne abhi prove ki hain.

$\csc (90°-\theta )=\frac{1}{\sin (90°-\theta )}=\frac{1}{\cos \theta }=\sec \theta$

$\sec (90°-\theta )=\frac{1}{\cos (90°-\theta )}=\frac{1}{\sin \theta }=\csc \theta$

$\cot (90°-\theta )=\frac{1}{\tan (90°-\theta )}=\frac{1}{\cot \theta }=\tan \theta$

Yeh step kyun? Cosecant, secant, aur cotangent respectively sine, cosine, aur tangent ke reciprocals hain. Identities ke dono sides par reciprocals apply karo.

Worked Examples

Diya hai: Hum jaante hain cos(30°) = √3/2.

Nikalna hai: Calculator ya table ke bina sin(60°).

Solution:

Dekho ki 60° = 90° − 30°.

Cofunction identity apply karo: $\sin (60°)=\sin (90°-30°)=\cos (30°)=\frac{\sqrt{3}}{2}$

Yeh step kyun? Humne identity sin(90° − θ) = cos θ use ki jisme θ = 30°.

Answer: sin(60°) = √3/2

Verification: Tum ise standard 30-60-90 triangle (sides 1: √3 : 2) se confirm kar sakte ho. 60° angle ke liye, opposite = √3, hypotenuse = 2, isliye sin(60°) = √3/2. ✓

Diya hai: Expression tan(90° − 25°).

Simplify: Cofunction identity use karo.

Solution:

Identity tan(90° − θ) = cot θ apply karo jisme θ = 25°: $\tan (90°-25°)=\cot (25°)$

Yeh step kyun? Tangent-cotangent cofunction pair ka direct application.

Hum ise yun bhi likh sakte hain: $\cot (25°)=\frac{1}{\tan (25°)}$

Yeh step kyun? Cotangent, tangent ka reciprocal hai, jo context ke hisaab se zyada useful ho sakta hai.

Answer: tan(65°) = cot(25°) = 1/tan(25°)

Diya hai: Expression sec(90° − A) · sin(A).

Prove karna hai: Yeh sabhi valid A ke liye 1 ke barabar hai.

Solution:

Step 1: Secant ke liye cofunction identity apply karo. $\sec (90°-A)=\csc (A)$

Yeh step kyun? Hum sec(90° − θ) = csc θ use karte hain jisme θ = A.

Step 2: Original expression mein substitute karo. $\sec (90°-A)\cdot \sin (A)=\csc (A)\cdot \sin (A)$

Step 3: Cosecant ke liye reciprocal identity use karo. $\csc (A)=\frac{1}{\sin (A)}$

Yeh step kyun? Cosecant define hota hai 1/sin se.

Step 4: Product simplify karo. $\csc (A)\cdot \sin (A)=\frac{1}{\sin (A)}\cdot \sin (A)=1$

Yeh step kyun? Koi bhi non-zero number apne reciprocal se multiply hone par 1 deta hai.

Answer: sec(90° − A) · sin(A) = 1 (proved) ∎

Diya hai: sin(3x) = cos(2x), aur hume x chahiye range 0° < x < 90° mein.

Nikalna hai: x ki value.

Solution:

Step 1: Cofunction identity use karo: cos(2x) = sin(90° − 2x). $\sin (3x)=\sin (90°-2x)$

Yeh step kyun? Hum cosine ko sine mein convert karte hain cofunction identity use karke taaki dono sides mein same function ho.

Step 2: Arguments ko equal set karo (acute angles ke liye). $3x=90°-2x$

Yeh step kyun? Agar sin(α) = sin(β) aur dono angles acute hain, to α = β (principal domain mein).

Step 3: x ke liye solve karo. $3x+2x=90°$ $5x=90°$ $x=18°$

Yeh step kyun? Basic algebra: like terms collect karo aur divide karo.

Answer: x = 18°

Verification: sin(54°) = sin(90° − 36°) = cos(36°). Check karo: 3(18°) = 54° aur 2(18°) = 36°. ✓

Common Mistakes

Kyun sahi lagta hai: "Complementary" aur "supplementary" dono similar sound karte hain aur angle pairs involve karte hain.

Fix yeh hai:

• Complementary angles 90° mein add hote hain (right angle).
• Supplementary angles 180° mein add hote hain (straight line).
• Cofunction identities sirf 90° (complementary) ke liye kaam karti hain, 180° ke liye nahi.
• Supplementary angles ke liye, sin(180° − θ) = sin θ hota hai, cos θ nahi!

Steel-man: Confusion natural hai kyunki dono standard angle-pair relationships hain. Mnemonic "C for Corner (90° ek corner/right angle hai)" inhe distinguish karne mein help karta hai.

Kyun sahi lagta hai: Identity same lagti hai, aur hum bhool jaate hain ki 90° ≠ π/2 jab θ galat unit mein ho.

Fix yeh hai:

• Agar θ degrees mein hai, use karo sin(90° − θ) = cos θ
• Agar θ radians mein hai, use karo sin(π/2 − θ) = cos θ
• Ek hi expression mein units kabhi mix mat karo!

Example: Agar θ = π/6 (radians), to sin(π/2 − π/6) = sin(π/3) = cos(π/6) = √3/2. Lekin agar tumne sin(90° − π/6) likha, to yeh nonsense hai (ek radian measure ko degree measure se subtract karna).

Kyun sahi lagta hai: Humne arithmetic compute ki (90 − 50 = 40) lekin identity use nahi ki.

Fix yeh hai:

• Pehle pattern recognize karo: "Yeh 90° minus kuch hai."
• Cofunction identity apply karo: sin(90° − 50°) = cos(50°).
• Tabhi evaluate karo agar zaroorat ho.

Yeh kyun matter karta hai: Algebra problems mein, tumhe aksar form cos(50°) chahiye hoti hai doosre terms match karne ke liye, numerical value sin(40°) ≈ 0.643 nahi.

80/20 Principle — Core to Master

Woh 20% jo 80% value deta hai:

1. Teen main pairs: sin ↔ cos, tan ↔ cot, sec ↔ csc jab angles complementary hain (90° mein add hote hain).
2. Quick recognition: Agar tum (90° − θ) ya (π/2 − θ) dekho, mentally cofunction mein swap karo.
3. Equations mein application: Agar sin(A) = cos(B), to A aur B complementary hain (A + B = 90°).

Yeh teen patterns master karo, aur tum cofunction identities involve karne wale 80% problems solve kar sakte ho.

Memory Aids

Har "co-" function apne non-"co-" partner ke saath pair karta hai jab angles complementary hon.

Active Recall Practice

Ab, sine aur cosine triangle ki sides compare karne ke fancy tarike hain. Cool part yeh hai: jab tum 30° angle ka sine measure karte ho, tum ek side ko longest side se compare kar rahe ho. Lekin jab tum 60° angle (doosra buddy angle) ka cosine measure karte ho, tum exactly usi side ko same longest side se dekh rahe ho! Isliye sine of 30° = cosine of 60°. Dono same number hain!

Yeh aisa hai jaise tum aur tumhara dost ek hi stick ko opposite ends se dekh rahe ho — tum ise "left side" keh sakte ho aur tumhara dost ise "right side" kehta hai, lekin hai wahi stick! Sine aur cosine roles swap kar lete hain jab angles complementary buddies hon. Isliye sin(90° − θ) = cos(θ) — tum bas triangle ko doosre angle ke point of view se dekh rahe ho.

Connections

• Right Triangle Trigonometry — woh foundation jahan se opposite/adjacent definitions aati hain
• Unit Circle — cofunction identities y = x line ke across reflections ki tarah dikhti hain
• Trigonometric Identities — cofunction identities broader identity family ka ek subset hain
• Angle Sum Formulas — cofunction identities derive ki ja sakti hain sin(90° − θ) = sin(90°)cos(θ) − cos(90°)sin(θ) use karke
• Even and Odd Functions — related symmetry concepts, lekin cofunctions complementary symmetry dikhate hain
• Solving Trigonometric Equations — cofunction identities aksar equations simplify karti hain functions ko unify karke
• Graph Transformations — sin(x) ko π/2 se left shift karne par cos(x) milta hai, jo phase relationship illustrate karta hai

#flashcards/maths

What are complementary angles? :: Do angles jo 90° (ya π/2 radians) mein sum karte hain.

Simplify: tan(90° − 40°) :: cot(40°) ya 1/tan(40°) ya tan(50°)

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps