Concept Core
Ratios, identities, compound angles, properties of triangles β the complete map.
Trigonometric Ratios & Fundamental Identity
For angle ΞΈ in a right triangle: sin ΞΈ = opp/hyp, cos ΞΈ = adj/hyp, tan ΞΈ = sin/cos = opp/adj.
sinΒ²ΞΈ + cosΒ²ΞΈ = 1 1 + tanΒ²ΞΈ = secΒ²ΞΈ 1 + cotΒ²ΞΈ = cosecΒ²ΞΈ
Signs by quadrant (ASTC): All (+) in Q1, Sine (+) in Q2, Tan (+) in Q3, Cos (+) in Q4.
Compound Angle Formulas
sin(AΒ±B) = sinA cosB Β± cosA sinB
cos(AΒ±B) = cosA cosB β sinA sinB
tan(AΒ±B) = (tanA Β± tanB)/(1 β tanA tanB)
Double Angle & Half Angle
sin 2A = 2 sinA cosA = 2tanA/(1+tanΒ²A)
cos 2A = cosΒ²AβsinΒ²A = 1β2sinΒ²A = 2cosΒ²Aβ1
tan 2A = 2tanA/(1βtanΒ²A)
sinΒ²A = (1βcos2A)/2; cosΒ²A = (1+cos2A)/2
Sum-to-Product Transformations
sinC + sinD = 2 sin((C+D)/2) cos((CβD)/2)
sinC β sinD = 2 cos((C+D)/2) sin((CβD)/2)
cosC + cosD = 2 cos((C+D)/2) cos((CβD)/2)
cosC β cosD = β2 sin((C+D)/2) sin((CβD)/2)
Product-to-Sum Transformations
2 sinA cosB = sin(A+B) + sin(AβB)
2 cosA cosB = cos(A+B) + cos(AβB)
2 sinA sinB = cos(AβB) β cos(A+B)
Properties of Triangles (Sine & Cosine Rule)
For triangle with sides a, b, c opposite to angles A, B, C:
Sine rule: a/sinA = b/sinB = c/sinC = 2R
Cosine rule: aΒ² = bΒ² + cΒ² β 2bc cosA
Area = Β½ ab sinC = β(s(sβa)(sβb)(sβc))
s = (a+b+c)/2 (semi-perimeter). R = circumradius.
Standard Values β The Trig Table You Must Know
| ΞΈ | sin | cos | tan |
|---|
| 0Β° | 0 | 1 | 0 |
| 30Β° | 1/2 | β3/2 | 1/β3 |
| 45Β° | 1/β2 | 1/β2 | 1 |
| 60Β° | β3/2 | 1/2 | β3 |
| 90Β° | 1 | 0 | undefined |
Formula Vault
Every trigonometric formula β identities, transformations, inverse, and triangle.
Pythagorean Identities
sinΒ²ΞΈ+cosΒ²ΞΈ=1
1+tanΒ²ΞΈ=secΒ²ΞΈ
1+cotΒ²ΞΈ=cosecΒ²ΞΈ
Fundamental; derive all others
sin(A+B)
sinA cosB + cosA sinB
Expand then simplify
cos(A+B)
cosA cosB β sinA sinB
Note minus sign for +
sin 2A
2 sinA cosA
Double angle
cos 2A
1β2sinΒ²A = 2cosΒ²Aβ1
Three equivalent forms
tan 2A
2tanA/(1βtanΒ²A)
Undefined when tanA = Β±1
sinC + sinD
2sin((C+D)/2)cos((CβD)/2)
Sum to product
Sine Rule
a/sinA = b/sinB = c/sinC = 2R
R = circumradius
Cosine Rule
aΒ² = bΒ²+cΒ²β2bc cosA
Use when 2 sides + included angle
Triangle Area
Ξ = Β½ab sinC = β(s(sβa)(sβb)(sβc))
Heron's formula; s = (a+b+c)/2
Worked Examples
5 problems β identity proof, compound angles, triangle, and a classic EAPCET trap.
EasyEvaluate sin 75Β°βΎ
Find the exact value of sin 75Β°.
1
sin 75Β° = sin(45Β° + 30Β°) = sin45Β°cos30Β° + cos45Β°sin30Β°
2
= (1/β2)(β3/2) + (1/β2)(1/2) = β3/(2β2) + 1/(2β2) = (β3+1)/(2β2)
3
= (β3+1)/(2β2) Γ β2/β2 = (β6+β2)/4
β sin 75Β° = (β6+β2)/4
EasyProve: sinΒ²A β sinΒ²B = sin(A+B)sin(AβB)βΎ
Prove the identity: sinΒ²A β sinΒ²B = sin(A+B)sin(AβB).
1
RHS = (sinA cosB + cosA sinB)(sinA cosB β cosA sinB)
2
= sinΒ²A cosΒ²B β cosΒ²A sinΒ²B
3
= sinΒ²A(1βsinΒ²B) β (1βsinΒ²A)sinΒ²B
4
= sinΒ²A β sinΒ²A sinΒ²B β sinΒ²B + sinΒ²A sinΒ²B
5
= sinΒ²A β sinΒ²B = LHS β
β Identity proved
MediumIf tanA = Β½, tanB = β
, find tan(A+B)βΎ
Given tan A = 1/2 and tan B = 1/3, find tan(A+B).
1
tan(A+B) = (tanA + tanB)/(1 β tanA tanB)
2
= (1/2 + 1/3)/(1 β (1/2)(1/3))
3
= (5/6)/(1 β 1/6) = (5/6)/(5/6) = 1
4
Since tan(A+B) = 1 β A+B = 45Β°
β tan(A+B) = 1, so A+B = 45Β°
EAPCET LevelIn triangle ABC, a=5, b=6, C=60Β°. Find c.βΎ
In triangle ABC, a = 5, b = 6, angle C = 60Β°. Find the side c.
1
Apply cosine rule: cΒ² = aΒ² + bΒ² β 2ab cosC
2
cΒ² = 25 + 36 β 2(5)(6)cos60Β° = 61 β 60(1/2) = 61 β 30 = 31
β c = β31
Trap Questiontan(A+B) = (tanA + tanB)/(1 β tanA tanB) β when is this undefined?βΎ
For A = 30Β°, B = 60Β°, find tan(A+B) using the formula. β οΈ Students apply blindly without checking.
1
tanA = 1/β3, tanB = β3. Product = (1/β3)(β3) = 1.
2
Formula: tan(A+B) = (tanA+tanB)/(1βtanA tanB) = (1/β3 + β3)/(1β1)
3
Denominator = 0 β formula is undefined
4
Geometrically: A+B = 30Β°+60Β° = 90Β°, and tan90Β° is undefined. The formula correctly fails here.
5
Always check if tanAΓtanB = 1 before applying the formula.
β tan(A+B) is undefined because A+B = 90Β° (denominator = 0)
Mistake DNA
5 trigonometry errors that EAPCET candidates repeatedly make.
β
sin(A+B) β sinA + sinB
This is the most common algebraic error in trigonometry. Trig functions don't distribute over addition.
β Wrong
sin(30Β°+60Β°) = sin30Β°+sin60Β°
= 0.5+0.866 = 1.366 β
(sin 90Β° = 1, not 1.366)
β Correct
sin(A+B) = sinAcosB + cosAsinB β
sin90Β° = sin30Β°cos60Β°
+cos30Β°sin60Β° = 1 β
sin(A+B) requires the compound angle formula. This error also occurs with cos, tan, β(a+b), log(a+b) β none of these distribute.
π
cos 2A Formula β Wrong Form for the Context
cos 2A has three equivalent forms. Using the wrong one makes the algebra much harder.
β Wrong
Proving sinΒ² identity:
Using cos2A=cosΒ²AβsinΒ²A
β messy manipulation β
β Correct
Use cos2A = 1β2sinΒ²A
β sinΒ²A = (1βcos2A)/2 β
Choose the form that
contains what you need
Choose the form of cos 2A based on what you want to isolate. Need sinΒ²A? Use 1β2sinΒ²A. Need cosΒ²A? Use 2cosΒ²Aβ1. Need both? Use cosΒ²AβsinΒ²A.
π
ASTC Sign Errors for Angles in Q2, Q3, Q4
Forgetting that cos and tan are negative in Q2, that sin and cos are negative in Q3.
β Wrong
cos 120Β° = cos 60Β° = Β½ β
(cos is negative in Q2)
β Correct
120Β° is in Q2: only sin+
cos 120Β° = βcos60Β° = βΒ½ β
ASTCTrick: All Sin Tan Cos
ASTC (Anti-clockwise): All positive Q1, Sin+ Q2, Tan+ Q3, Cos+ Q4. For supplementary angles: sin(180Β°βΞΈ) = sinΞΈ, cos(180Β°βΞΈ) = βcosΞΈ.
π
Sine Rule: Using Wrong Angle-Side Pairing
In a/sinA = b/sinB, side a must be opposite to angle A. Mismatching sides and angles gives wrong answers.
β Wrong
a=5 opposite to angle B:
a/sinA = 5/sinA β
(should be a/sinB if
a is opposite B)
β Correct
a opposite A, b opposite B β
a/sinA = b/sinB β
Draw the triangle to
verify the pairing
In sine rule, each fraction is (side opposite angle)/(sin of that angle). Label your triangle clearly before setting up the ratio.
π
Quadrant Confusion with tanβ»ΒΉ: Forgetting Range
tanβ»ΒΉ returns values only in (β90Β°, 90Β°). Angles in Q2 and Q3 need adjustment.
β Wrong
tan ΞΈ = β1, Q2:
ΞΈ = tanβ»ΒΉ(β1) = β45Β° β
(β45Β° is in Q4, not Q2)
β Correct
tanβ»ΒΉ(β1) = β45Β° is reference
Q2: ΞΈ = 180Β° + (β45Β°) = 135Β° β
Or: ΞΈ = 180Β° β 45Β° = 135Β° β
The tanβ»ΒΉ function only gives the principal value in (β90Β°, 90Β°). Always check which quadrant ΞΈ is in from the original equation, then adjust.
Chapter Intelligence
Trigonometry is the single highest-weightage chapter in EAPCET Mathematics.
EAPCET Weightage (2019β2024)
Compound angles & identities~9 Properties of triangles~7 Sum-to-product / product-to-sum~5 Trigonometric equations~3 High-Yield PYQ Patterns
Evaluate sin/cos of 75Β°, 15Β°, etc.Prove identity using double anglesIf tanA=x, find sin2A or cos2ACosine rule: find unknown sideTriangle area using Heron's formulaSum-to-product simplificationtan(A+B) given tanA and tanB
Exam Strategy
- Memorise the compound angle formulas perfectly. Hundreds of identities can be derived from just sin(A+B) and cos(A+B) β derive, don't memorise all.
- When stuck on an identity, convert everything to sin and cos. This almost always makes the path forward visible.
- For triangle problems: if two sides + included angle β Cosine Rule. If two angles + one side β Sine Rule. Area β Β½ab sinC.
- ASTC mnemonic for signs: All Students Take Calculus (Q1: All, Q2: Sin, Q3: Tan, Q4: Cos).
- Trigonometry links to Coordinate Geometry (angle of inclination), Vectors (dot product = ab cosΞΈ), and Complex Numbers (polar form). This chapter's time investment pays off everywhere.
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