Trigonometry — vidhyapath 2027

Concept Core

Ratios, identities, compound angles, properties of triangles — the complete map.

Trigonometric Ratios & Fundamental Identity

For angle θ in a right triangle: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = sin/cos = opp/adj.

sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ

Signs by quadrant (ASTC): All (+) in Q1, Sine (+) in Q2, Tan (+) in Q3, Cos (+) in Q4.

Compound Angle Formulas

sin(A±B) = sinA cosB ± cosA sinB cos(A±B) = cosA cosB ∓ sinA sinB tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)

Double Angle & Half Angle

sin 2A = 2 sinA cosA = 2tanA/(1+tan²A) cos 2A = cos²A−sin²A = 1−2sin²A = 2cos²A−1 tan 2A = 2tanA/(1−tan²A) sin²A = (1−cos2A)/2; cos²A = (1+cos2A)/2

Sum-to-Product Transformations

sinC + sinD = 2 sin((C+D)/2) cos((C−D)/2) sinC − sinD = 2 cos((C+D)/2) sin((C−D)/2) cosC + cosD = 2 cos((C+D)/2) cos((C−D)/2) cosC − cosD = −2 sin((C+D)/2) sin((C−D)/2)

Product-to-Sum Transformations

2 sinA cosB = sin(A+B) + sin(A−B) 2 cosA cosB = cos(A+B) + cos(A−B) 2 sinA sinB = cos(A−B) − 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: a² = b² + c² − 2bc cosA Area = ½ ab sinC = √(s(s−a)(s−b)(s−c))

s = (a+b+c)/2 (semi-perimeter). R = circumradius.

Standard Values — The Trig Table You Must Know

θ	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Formula Vault

Every trigonometric formula — identities, transformations, inverse, and triangle.

Pythagorean Identities

sin²θ+cos²θ=1
1+tan²θ=sec²θ
1+cot²θ=cosec²θ

Fundamental; derive all others

sin(A+B)

sinA cosB + cosA sinB

Expand then simplify

cos(A+B)

cosA cosB − sinA sinB

Note minus sign for +

sin 2A

2 sinA cosA

Double angle

cos 2A

1−2sin²A = 2cos²A−1

Three equivalent forms

tan 2A

2tanA/(1−tan²A)

Undefined when tanA = ±1

sinC + sinD

2sin((C+D)/2)cos((C−D)/2)

Sum to product

Sine Rule

a/sinA = b/sinB = c/sinC = 2R

R = circumradius

Cosine Rule

a² = b²+c²−2bc cosA

Use when 2 sides + included angle

Triangle Area

Δ = ½ab sinC = √(s(s−a)(s−b)(s−c))

Heron's formula; s = (a+b+c)/2

Worked Examples

5 problems — identity proof, compound angles, triangle, and a classic EAPCET trap.

EasyEvaluate sin 75°▾

Find the exact value of sin 75°.

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√2) × √2/√2 = (√6+√2)/4

✓ sin 75° = (√6+√2)/4

EasyProve: sin²A − sin²B = sin(A+B)sin(A−B)▾

Prove the identity: sin²A − sin²B = sin(A+B)sin(A−B).

RHS = (sinA cosB + cosA sinB)(sinA cosB − cosA sinB)

= sin²A cos²B − cos²A sin²B

= sin²A(1−sin²B) − (1−sin²A)sin²B

= sin²A − sin²A sin²B − sin²B + sin²A sin²B

= sin²A − sin²B = LHS ✓

✓ Identity proved

MediumIf tanA = ½, tanB = ⅓, find tan(A+B)▾

Given tan A = 1/2 and tan B = 1/3, find tan(A+B).

tan(A+B) = (tanA + tanB)/(1 − tanA tanB)

= (1/2 + 1/3)/(1 − (1/2)(1/3))

= (5/6)/(1 − 1/6) = (5/6)/(5/6) = 1

Since tan(A+B) = 1 → A+B = 45°

✓ tan(A+B) = 1, so A+B = 45°

EAPCET LevelIn triangle ABC, a=5, b=6, C=60°. Find c.▾

In triangle ABC, a = 5, b = 6, angle C = 60°. Find the side c.

Apply cosine rule: c² = a² + b² − 2ab cosC

c² = 25 + 36 − 2(5)(6)cos60° = 61 − 60(1/2) = 61 − 30 = 31

c = √31

✓ c = √31

Trap Questiontan(A+B) = (tanA + tanB)/(1 − tanA tanB) — when is this undefined?▾

For A = 30°, B = 60°, find tan(A+B) using the formula. ⚠️ Students apply blindly without checking.

tanA = 1/√3, tanB = √3. Product = (1/√3)(√3) = 1.

Formula: tan(A+B) = (tanA+tanB)/(1−tanA tanB) = (1/√3 + √3)/(1−1)

Denominator = 0 → formula is undefined

Geometrically: A+B = 30°+60° = 90°, and tan90° is undefined. The formula correctly fails here.

Always check if tanA×tanB = 1 before applying the formula.

✓ tan(A+B) is undefined because A+B = 90° (denominator = 0)

Mistake DNA

5 trigonometry errors that EAPCET candidates repeatedly make.

➕

sin(A+B) ≠ sinA + sinB

This is the most common algebraic error in trigonometry. Trig functions don't distribute over addition.

❌ Wrong

sin(30°+60°) = sin30°+sin60° = 0.5+0.866 = 1.366 ✗ (sin 90° = 1, not 1.366)

✓ Correct

sin(A+B) = sinAcosB + cosAsinB ✓ sin90° = sin30°cos60° +cos30°sin60° = 1 ✓

sin(A+B) requires the compound angle formula. This error also occurs with cos, tan, √(a+b), log(a+b) — none of these distribute.

🔄

cos 2A Formula — Wrong Form for the Context

cos 2A has three equivalent forms. Using the wrong one makes the algebra much harder.

❌ Wrong

Proving sin² identity: Using cos2A=cos²A−sin²A → messy manipulation ✗

✓ Correct

Use cos2A = 1−2sin²A → sin²A = (1−cos2A)/2 ✓ Choose the form that contains what you need

Choose the form of cos 2A based on what you want to isolate. Need sin²A? Use 1−2sin²A. Need cos²A? Use 2cos²A−1. Need both? Use cos²A−sin²A.

📐

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.

❌ Wrong

cos 120° = cos 60° = ½ ✗ (cos is negative in Q2)

✓ Correct

120° is in Q2: only sin+ cos 120° = −cos60° = −½ ✓ ASTCTrick: All Sin Tan Cos

ASTC (Anti-clockwise): All positive Q1, Sin+ Q2, Tan+ Q3, Cos+ Q4. For supplementary angles: sin(180°−θ) = sinθ, cos(180°−θ) = −cosθ.

📏

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.

❌ Wrong

a=5 opposite to angle B: a/sinA = 5/sinA ✗ (should be a/sinB if a is opposite B)

✓ Correct

a opposite A, b opposite B ✓ a/sinA = b/sinB ✓ 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.

🌀

Quadrant Confusion with tan⁻¹: Forgetting Range

tan⁻¹ returns values only in (−90°, 90°). Angles in Q2 and Q3 need adjustment.

❌ Wrong

tan θ = −1, Q2: θ = tan⁻¹(−1) = −45° ✗ (−45° is in Q4, not Q2)

✓ Correct

tan⁻¹(−1) = −45° is reference Q2: θ = 180° + (−45°) = 135° ✓ Or: θ = 180° − 45° = 135° ✓

The tan⁻¹ function only gives the principal value in (−90°, 90°). Always check which quadrant θ is in from the original equation, then adjust.

Chapter Intelligence

Trigonometry is the single highest-weightage chapter in EAPCET Mathematics.

EAPCET Weightage (2019–2024)

Compound angles & identities

Properties of triangles

Double angle formulas

Sum-to-product / product-to-sum

Inverse trig functions

Trigonometric equations

High-Yield PYQ Patterns

Evaluate sin/cos of 75°, 15°, 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) — 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 → Cosine Rule. If two angles + one side → Sine Rule. Area → ½ab 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 cosθ), and Complex Numbers (polar form). This chapter's time investment pays off everywhere.