Trigonometry (త్రికోణమితి)

sin θ=5/13, cos θ=12/13, tan θ=5/12

SOH-CAH-TOA:
sin θ = opp/hyp = 5/13.
cos θ = adj/hyp = 12/13.
tan θ = opp/adj = 5/12.

Verify: sin²+cos²=25/169+144/169=169/169=1 ✓
5-12-13 is a Pythagorean triple — exam favourite!

LHS = sin²θ/cos²θ - sin²θ = sin²θ(1-cos²θ)/cos²θ = sin²θ×sin²θ/cos²θ = tan²θ×sin²θ = RHS ✓

LHS = tan²θ - sin²θ.
= sin²θ/cos²θ - sin²θ.
= sin²θ(1/cos²θ - 1).
= sin²θ × (1-cos²θ)/cos²θ.
= sin²θ × sin²θ/cos²θ.
= sin²θ × tan²θ = RHS ✓

θ = 90°, 210°, 330°

Let s=sinθ: 2s²-s-1=0 → (2s+1)(s-1)=0.
s=1: sinθ=1 → θ=90°.
s=-1/2: sinθ=-1/2 → θ=210° or 330°.

sinθ=-1/2 in Q3 and Q4 (sin negative):
Q3: 180°+30°=210°. Q4: 360°-30°=330°.

sin 30° = 1/2

Standard values గుర్తు పెట్టుకోండి:
sin 0°=0, sin 30°=1/2, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1.

Trick: 0,30,45,60,90 కు sin values = √0/2, √1/2, √2/2, √3/2, √4/2.
= 0, 1/2, 1/√2, √3/2, 1. ఇలా remember చేయడం easy!

tan(A+B) = C/(1-D)

tan(A+B) = (tanA+tanB)/(1-tanA×tanB) = C/(1-D).

sin 60° = 2×(1/2)×(√3/2) = √3/2 ✓

LHS: sin 2×30° = sin 60° = √3/2.
RHS: 2×sin 30°×cos 30° = 2×(1/2)×(√3/2) = √3/2.
LHS = RHS ✓.

sin 2θ = 2sinθcosθ — Double Angle Formula!

sec 60° = 2

sec θ = 1/cos θ.
sec 60° = 1/cos 60° = 1/(1/2) = 2.

గుర్తు: sec values: 1, 2/√3, √2, 2, undefined (0°,30°,45°,60°,90°).
sec 60° = 2 అంటే hypotenuse = 2 × adjacent at 60°!

= 1

Pair up: tan θ × tan(90°-θ) = tanθ × cotθ = 1.
Pairs: (1°,89°), (2°,88°), ..., (44°,46°).
Each pair product = 1. Middle: tan 45° = 1.
Total = 1×1×...×1 = 1.

= 1/8

Formula: cosθ×cos(60°-θ)×cos(60°+θ) = (1/4)cos3θ.
θ=20°: cos20°×cos40°×cos80°.
cos(60°-20°)=cos40°, cos(60°+20°)=cos80° ✓.
= (1/4)cos60° = (1/4)(1/2) = 1/8.

Maximum = √(a²+b²)

asinθ + bcosθ = √(a²+b²) × sin(θ+φ) where tanφ=b/a.
Maximum of sin = 1.
Maximum = √(a²+b²).

ఉదా: 3sinθ+4cosθ max = √(9+16) = 5.
Minimum = -√(a²+b²).

sin θ=5/13, cos θ=12/13

tan=opp/adj=5/12 → opp=5,adj=12.
hyp=√(25+144)=√169=13.
sin=5/13, cos=12/13.

5-12-13 Pythagorean triple!

cos 30° = √3/2

30°-60°-90° triangle sides: 1, √3, 2.
cos 30° = adj/hyp = √3/2 ≈ 0.866.

గుర్తు: cos 30° = sin 60° = √3/2 (complementary angles!).
Larger angle → larger opposite → larger sin, smaller cos.

cos θ = √(b²-a²)/b

cos²θ = 1-sin²θ = 1-a²/b² = (b²-a²)/b².
cos θ = √(b²-a²)/b.

ఉదా: a=3, b=5 → cos θ=√(25-9)/5=4/5. (3-4-5!)

= 5π/6

cos(7π/6) = cos(π+π/6) = -cosπ/6 = -√3/2.
cos⁻¹(-√3/2) = 5π/6 (principal value range [0,π]).

cos⁻¹ range = [0,π]. 7π/6 ∉ [0,π].
cos(5π/6) = -√3/2 ✓, and 5π/6 ∈ [0,π] ✓.

cos 15° = (√6+√2)/4

cos 15° = cos(45°-30°) = cos45°cos30° + sin45°sin30°.
= (1/√2)(√3/2) + (1/√2)(1/2).
= √3/(2√2) + 1/(2√2) = (√3+1)/(2√2).
= (√3+1)×√2/4 = (√6+√2)/4.

cos 15° = sin 75° = (√6+√2)/4 ≈ 0.966.

cos(A-B)=cosAcosB+sinAsinB; cos(A+B)=cosAcosB-sinAsinB. Subtract→2sinAsinB=cos(A-B)-cos(A+B) ✓

RHS = ½[cos(A-B)-cos(A+B)].
cos(A-B) = cosAcosB+sinAsinB.
cos(A+B) = cosAcosB-sinAsinB.
Difference = 2sinAsinB.
½×2sinAsinB = sinAsinB = LHS ✓.

C=π-A-B. Use double angle, sum formulas step by step

sin²A+sin²B+sin²C.
= (1-cos2A)/2+(1-cos2B)/2+(1-cos2C)/2.
= 3/2 - (cos2A+cos2B+cos2C)/2.
cos2A+cos2B+cos2C = -1-4cosAcosBcosC (known identity for A+B+C=π).
= 3/2-(-1-4cosAcosBcosC)/2 = 3/2+1/2+2cosAcosBcosC = 2+2cosAcosBcosC.
= 2(1+cosAcosBcosC) ✓.

tan θ = sin θ / cos θ

tan θ = (opp/hyp) ÷ (adj/hyp) = opp/adj = sin θ/cos θ.

గుర్తు: tan = "tangent" — line tangent to unit circle!
Also: cot θ = cos θ/sin θ = 1/tan θ.

0 + 0 = 0

sin 0° = 0.
cos 90° = 0.
Sum = 0 + 0 = 0.

గుర్తు: sin 0°=0 (no height at 0°), cos 90°=0 (no horizontal at 90°).
These are boundary values — very commonly tested!

tan 0° = 0; tan 90° = undefined (∞)

tan 0° = sin 0°/cos 0° = 0/1 = 0.
tan 90° = sin 90°/cos 90° = 1/0 = undefined (infinity).

గుర్తు: Division by zero → undefined!
tan 90° ని "infinity" అంటాం — angle vertical అయినపుడు slope infinite.

sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC

sin(A+B+C) = sin[(A+B)+C].
= sin(A+B)cosC + cos(A+B)sinC.
= (sinAcosB+cosAsinB)cosC + (cosAcosB-sinAsinB)sinC.
= sinAcosBcosC+cosAsinBcosC+cosAcosBsinC-sinAsinBsinC.

1 + tan²θ = sec²θ

Pythagorean identities (3 versions):
1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ (divide identity 1 by cos²θ)
3. 1 + cot²θ = cosec²θ (divide identity 1 by sin²θ)

గుర్తు: sin²+cos²=1 నుండి మిగతా రెండు derive చేయవచ్చు!

cosec θ × sin θ = 1

cosec θ = 1/sin θ.
cosec θ × sin θ = (1/sin θ) × sin θ = 1.

Similarly: sec θ × cos θ = 1, tan θ × cot θ = 1.
గుర్తు: Reciprocal pairs: sin-cosec, cos-sec, tan-cot. Product always = 1!

cos(A+B)=cosAcosB-sinAsinB. Let A=cos⁻¹x, B=cos⁻¹y → cos(A+B)=xy-√(1-x²)√(1-y²) ✓

Let A=cos⁻¹x, B=cos⁻¹y.
cosA=x, cosB=y.
sinA=√(1-x²), sinB=√(1-y²).
cos(A+B)=cosAcosB-sinAsinB=xy-√(1-x²)√(1-y²).
A+B=cos⁻¹(xy-√(1-x²)√(1-y²)) ✓.

= 1 - 2sinθcosθ = 1 - sin2θ

(sinθ-cosθ)² = sin²θ - 2sinθcosθ + cos²θ.
= (sin²θ+cos²θ) - 2sinθcosθ.
= 1 - sin2θ.

Compare: (sinθ+cosθ)² = 1+sin2θ.
Difference = 4sinθcosθ = 2sin2θ.

sin(90°-θ) = cos θ

Complementary angle identities:
sin(90°-θ) = cos θ.
cos(90°-θ) = sin θ.
tan(90°-θ) = cot θ.
cot(90°-θ) = tan θ.

గుర్తు: 90° minus చేస్తే sin↔cos, tan↔cot, sec↔cosec swap అవుతాయి!
ఉదా: sin 30° = cos 60° = 1/2 ✓

θ = 45°

sinθ = cosθ → sinθ/cosθ = 1 → tanθ = 1 → θ = 45°.

At θ=45°: sin45°=cos45°=1/√2.
This is the only angle in (0°,90°) where sin=cos!

T₂(cosθ) = 2cos²θ-1 = cos2θ ✓

cos2θ = 2cos²θ-1.
Let x=cosθ: T₂(x) = 2x²-1.
T₂(cosθ) = 2cos²θ-1 = cos2θ ✓.

Chebyshev polynomials: T_n(cosθ) = cos(nθ).
T₁(x)=x, T₂(x)=2x²-1, T₃(x)=4x³-3x...

tan 30° = 1/√3

tan 30° = sin 30°/cos 30° = (1/2)/(√3/2) = 1/√3 = √3/3.

గుర్తు: tan values 0,30,45,60,90: 0, 1/√3, 1, √3, ∞.
Pattern: 0 < 1/√3 < 1 < √3 < ∞ (increasing!)

-30° లేదా 330°

sin 210° = sin(180°+30°) = -sin30° = -1/2.
sin⁻¹(-1/2) = -30° (principal value, range [-90°,90°]).

గుర్తు: sin⁻¹ range = [-π/2, π/2].
sin⁻¹(sin θ) = θ only if θ ∈ [-90°,90°].
Otherwise use the equivalent angle in range!

sin20°sin40°sin80° = (1/4)sin60° = √3/8

sinα×sin(60°+α)×sin(60°-α) = (1/4)sin3α.
α=20°: sin20°×sin80°×sin40° = (1/4)sin60° = (1/4)(√3/2) = √3/8.

sin60°+20°=sin80°, sin60°-20°=sin40° ✓.
Standard result: sin20°sin40°sin80° = √3/8!

(1+tanθ)/(1-tanθ)

Divide numerator and denominator by cosθ:
(cosθ+sinθ)/cosθ = 1+tanθ.
(cosθ-sinθ)/cosθ = 1-tanθ.
Result = (1+tanθ)/(1-tanθ) = tan(45°+θ).

ఉదా: θ=0°: (1+0)/(1-0)=1. θ=45°: undefined (tan90°).

cos 90° = 0

cos 2×45° = 1-2sin²45° = 1-2×(1/2) = 1-1 = 0.
cos 90° = 0 ✓.

Double angle formula: cos 2θ = 1-2sin²θ = 2cos²θ-1 = cos²θ-sin²θ.
All three are equivalent!

sin²θ + cos²θ = 1

ఇది Trigonometry లో అత్యంత ముఖ్యమైన identity!
Proof: Right triangle లో (opp² + adj²)/hyp² = hyp²/hyp² = 1.
sin²θ + cos²θ = 1 ఎల్లప్పుడూ valid — ఏ angle అయినా.

ఇందు నుండి: sin²θ = 1-cos²θ, cos²θ = 1-sin²θ.

Let P=A+B, Q=A-B. cos(A+B)+cos(A-B)=2cosAcosB → P,Q substitution ✓

cos(A+B) = cosAcosB-sinAsinB.
cos(A-B) = cosAcosB+sinAsinB.
Sum = 2cosAcosB.
Let P=A+B, Q=A-B → A=(P+Q)/2, B=(P-Q)/2.
cosP+cosQ = 2cos((P+Q)/2)cos((P-Q)/2). ✓

cos²θ - sin²θ = cos 2θ

Double angle formula:
cos 2θ = cos²θ - sin²θ.
Also: cos 2θ = 2cos²θ - 1 = 1 - 2sin²θ.

ఉదా: cos²45°-sin²45°=1/2-1/2=0=cos 90° ✓.

= 2secθ

cosθ/(1-sinθ) + cosθ/(1+sinθ) = cosθ[(1+sinθ+1-sinθ)/((1-sinθ)(1+sinθ))].
= cosθ × 2/(1-sin²θ) = 2cosθ/cos²θ = 2/cosθ = 2secθ.

= 2

sinθ + 1/sinθ = 2 → sin²θ - 2sinθ + 1 = 0 → (sinθ-1)² = 0 → sinθ = 1.
sin⁹θ = 1⁹ = 1. cosec⁹θ = 1.
Sum = 2.

గుర్తు: sinθ+cosecθ=2 → sinθ=1 → θ=90° (only solution).

sin 75° = (√6+√2)/4

sin 75° = sin(45°+30°) = sin45°cos30° + cos45°sin30°.
= (1/√2)(√3/2) + (1/√2)(1/2).
= √3/(2√2) + 1/(2√2).
= (√3+1)/(2√2) = (√3+1)√2/4 = (√6+√2)/4.

sin 75° ≈ 0.966.

cosec θ = 1/sin θ = hypotenuse/opposite

Reciprocal trigonometric ratios:
cosec θ = 1/sin θ = hyp/opp.
sec θ = 1/cos θ = hyp/adj.
cot θ = 1/tan θ = adj/opp.

గుర్తు: cosec↔sin, sec↔cos, cot↔tan (pairs!).
"co" prefix swaps the ratio!

tan2θ = -4/3

tan2θ = 2tanθ/(1-tan²θ) = 2×2/(1-4) = 4/(-3) = -4/3.

Double angle: tan2θ = 2tanθ/(1-tan²θ).
Note: tan²θ=4>1 → 1-tan²θ<0 → tan2θ negative. This means 2θ is in Q2 or Q4!

= 1

sin²30° + cos²30° = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1.

sin²θ + cos²θ = 1 identity verified!
ఇది ఏ angle కైనా = 1. Exam లో ఈ identity remember చేయడం must!

Opposite = 6

sin θ = opp/hyp = 3/5.
opp = (3/5) × 10 = 6.

Check: adj=√(100-36)=8. sin θ=6/10=3/5 ✓, cos θ=8/10=4/5.
6-8-10 = 3-4-5 triple!

cosec 30° = 2

cosec θ = 1/sin θ.
cosec 30° = 1/sin 30° = 1/(1/2) = 2.

గుర్తు: cosec values are reciprocals of sin values.
sin 30°=1/2 → cosec 30°=2. Easy!

= π

tan⁻¹1 = π/4.
tan⁻¹2+tan⁻¹3: xy=6>1, so formula gives π+tan⁻¹((2+3)/(1-6))=π+tan⁻¹(-1)=π-π/4=3π/4.
Total = π/4+3π/4 = π.

Standard result: tan⁻¹1+tan⁻¹2+tan⁻¹3=π.

cos 90° = 0 ✓

cos(45°+45°) = cos45°cos45° - sin45°sin45°.
= (1/√2)(1/√2) - (1/√2)(1/√2).
= 1/2 - 1/2 = 0 ✓.

cos 90° = 0 confirmed by addition formula!

= (1/4)sin3θ

Using product formula:
sinθ × sin(60°-θ) × sin(60°+θ) = (1/4)sin3θ.

Proof: sin(60°-θ)×sin(60°+θ) = sin²60°-sin²θ = 3/4-sin²θ.
sinθ×(3/4-sin²θ) = (3sinθ-4sin³θ)/4 = sin3θ/4.

Similarly: cosθcos(60°-θ)cos(60°+θ) = (1/4)cos3θ.

sin θ = 1/√2

tanθ + cotθ = sinθ/cosθ + cosθ/sinθ = (sin²θ+cos²θ)/(sinθcosθ) = 1/(sinθcosθ) = 2.
sinθcosθ = 1/2.
sin2θ = 2sinθcosθ = 1 → 2θ = 90° → θ = 45°.
sinθ = sin45° = 1/√2.

= 1 (= sin 90°)

sin(A+B) = sinAcosB + cosAsinB.
= sin(30°+60°) = sin 90° = 1.

Or directly: (1/2)(1/2) + (√3/2)(√3/2) = 1/4 + 3/4 = 1 ✓.

tan θ = 4/3

3sinθ = 4cosθ → sinθ/cosθ = 4/3 → tanθ = 4/3.

గుర్తు: sinθ/cosθ = tanθ directly.
This implies 3-4-5 triangle: sinθ=4/5, cosθ=3/5.