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AFCAT Trig Functions & Identities

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This page covers AFCAT Trig Functions & Identities with complete concept notes, 39 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Trig Functions & Identities— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

Trigonometric functions and identities form the backbone of NDA Mathematics. They appear in 3-4 questions every year, making them absolutely crucial for your success. Core Concept: Trigonometric functions relate angles to ratios of sides in a right triangle. The six basic functions are sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). These functions have specific relationships called identities that remain true for all angle values.

2.Key Rules

Core rules you must know cold

The fundamental trigonometric ratios are: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The reciprocal functions are: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

3.Formula Block

Memorise — at least one formula appears in every paper

• sin²θ + cos²θ = 1

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

• sin(A ± B) = sinA cosB ± cosA sinB

• cos(A ± B) = cosA cosB ∓ sinA sinB

• tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)

• sin2θ = 2sinθ cosθ

• cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ

• tan2θ = 2tanθ/(1 - tan²θ)

Block:

Pythagorean Identities:

Sum and Difference Formulas:

Double Angle Formulas:

4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA consistently asks questions on identity verification, simplification of trigonometric expressions, and finding values of trigonometric functions at specific angles. Combination problems involving multiple identities are very common. Questions often test angles like 0°, 30°, 45°, 60°, 90° and their multiples. Powerful Shortcut: For quick calculations, remember the magic triangle for standard angles. For sin values: 0°=0/2, 30°=1/2, 45°=√2/2, 60°=√3/2, 90°=2/2.

For cos values, reverse this sequence. This eliminates lengthy calculations in MCQs.

5.Worked Example

Solve this step-by-step before moving on

1

Step 1

Use basic identities. We know 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ

2

Step 2

Substitute: sec²θ/cosec²θ

3

Step 3

Convert to basic functions: (1/cos²θ)/(1/sin²θ) = (1/cos²θ) × (sin²θ/1) = sin²θ/cos²θ

4

Step 4

Simplify: sin²θ/cos²θ = (sinθ/cosθ)² = tan²θ Therefore, the identity is proved. Worked Example 2: Find the value of sin15°

1

Step 1

Express 15° as difference of known angles: 15° = 45° - 30°

2

Step 2

Apply sin(A-B) formula: sin(45° - 30°) = sin45° cos30° - cos45° sin30°

3

Step 3

Substitute known values: (√2/2)(√3/2) - (√2/2)(1/2)

4

Step 4

Simplify: (√6/4) - (√2/4) = (√6 - √2)/4 Therefore, sin15° = (√6 - √2)/4 Another

6.Shortcuts

Use these to save 30–60 seconds per question

For complementary angles, remember sin(90° - θ) = cosθ and cos(90° - θ) = sinθ. This helps solve many problems instantly without lengthy calculations. Most

7.Exam Traps

Common mistakes students make — avoid these

Students frequently confuse the signs in sum and difference formulas. Remember the pattern: sine takes the SAME sign as the operation (+ or -), while cosine takes the OPPOSITE sign. For cos(A + B), use the minus sign; for cos(A - B), use the plus sign.

This single confusion costs many marks in NDA exams. Success Strategy: Master the standard angle values first, then focus on identity manipulation. Practice converting complex expressions to simpler forms using basic identities. Time management is crucial - if an identity proof seems lengthy, look for a shorter path using reciprocal relationships.

Key Points to Remember

sin²θ + cos²θ = 1 is the most fundamental identity - memorize this first
Quick ratio: sin30° = 1/2, sin45° = √2/2, sin60° = √3/2, cos values are reverse
Reciprocal trick: sinθ × cosecθ = 1, cosθ × secθ = 1, tanθ × cotθ = 1
Sum formula memory: sin(A+B) = sinA cosB + cosA sinB (same signs)
Difference formula: cos(A-B) = cosA cosB + sinA sinB (opposite to addition)
Double angle shortcut: sin2θ = 2sinθ cosθ, most frequently tested formula
Complementary angle rule: sin(90°-θ) = cosθ, cos(90°-θ) = sinθ
Pythagorean family: Three identities using sin²θ + cos²θ = 1 as base
Standard angles 0°, 30°, 45°, 60°, 90° appear in 80% of NDA trig questions
Identity verification problems can be solved by converting everything to sin and cos

Exam-Specific Tips

sin0° = 0, sin30° = 1/2, sin45° = √2/2, sin60° = √3/2, sin90° = 1
cos0° = 1, cos30° = √3/2, cos45° = √2/2, cos60° = 1/2, cos90° = 0
tan45° = 1, tan30° = 1/√3, tan60° = √3, tan0° = 0, tan90° = undefined
sec²θ - tan²θ = 1 for all values of θ where secθ and tanθ are defined
cosec²θ - cot²θ = 1 for all values of θ where cosecθ and cotθ are defined
sin²θ + cos²θ = 1 holds true for every real number θ
cos2θ has three equivalent forms: cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
tan(A+B) = (tanA + tanB)/(1 - tanA tanB) when denominator ≠ 0

Practice MCQs

Trig Functions & Identities — Practice Questions

39graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 39

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Practice 1easy

If sin θ = 3/5 and θ lies in the second quadrant, find the value of tan θ.

Practice 2easy

If sin θ = 3/5 and θ lies in the second quadrant, find the value of cos θ.

Practice 3easy

If cos A = 4/5 and sin B = 5/13, where A and B are acute angles, find sin(A + B).

Practice 4easy

Find the principal value of sin⁻¹(sin 5π/6).

Practice 5easy

Simplify: (sin 2θ) / (1 + cos 2θ)

Practice 6easy

Simplify: sin(60° + θ) cos(60° - θ) + cos(60° + θ) sin(60° - θ)

Practice 7easy

Simplify: sin²(π/8) + sin²(3π/8)

Practice 8easy

Find the principal value of sin⁻¹(sin(7π/6)).

Practice 9easy

If tan θ = 5/12 and θ is acute, find sin(2θ).

Practice 10easy

Simplify: cos²(π/8) - sin²(π/8)

Practice 11easy

Simplify: sin(60° + θ) + sin(60° − θ)

Practice 12easy

If tan θ = 5/12 and θ is acute, find sin 2θ.

Practice 13easy

Simplify: cos²(π/8) − sin²(π/8)

Practice 14medium

If sin(α + β) = 1 and sin(α − β) = 1/2, where α, β ∈ [0, π/2], then the value of tan(α)/tan(β) is:

Practice 15medium

If sin(α) + sin(β) = a and cos(α) + cos(β) = b, where a² + b² ≠ 0, then cos(α − β) is equal to:

Practice 16medium

If sin(A + B) = 1 and sin(A − B) = 1/2, where A, B ∈ [0, π/2], then the value of tan(A)/tan(B) is:

Practice 17medium

The value of cos⁻¹(cos(7π/6)) is:

Practice 18medium

If tan(θ/2) = t, then sin(θ) − cos(θ) equals:

Practice 19medium

If sin⁻¹(x) + sin⁻¹(y) = π/2, then the value of x² + y² is:

Practice 20medium

The value of sin(π/8) · sin(3π/8) · sin(5π/8) · sin(7π/8) is:

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60-Second Revision — Trig Functions & Identities

Formula: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
Remember: Standard angles - 0°, 30°, 45°, 60°, 90° values by heart
Trick: sin(90°-θ) = cosθ, cos(90°-θ) = sinθ for complementary angles
Pattern: sin(A±B) keeps same sign, cos(A±B) uses opposite sign
Trap: Don't confuse signs in sum/difference formulas - practice the pattern
Speed: Convert complex expressions to sin and cos for easier manipulation
Priority: Double angle formulas sin2θ and cos2θ are most frequently tested

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