How to Solve: Trigonometry Ratios and Identities

How to Solve: Trigonometry Ratios and Identities

Complete Guide for SSC / Bank / Railway Exams

Introduction

"Mastering trigonometry ratios and identities can add 5–10 marks to your SSC, Bank, or Railway exam—enough to push you from ‘just passing’ to ‘top rank.’ These questions appear in every paper, and if you follow this exact method, you’ll solve them in under 60 seconds."

What You Need To Know First

1. Basic right-triangle definitions of sine, cosine, and tangent.
2. Pythagorean theorem (a² + b² = c²).
3. Reciprocal identities (cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ).

Key Vocabulary

Formulas To Know

1. Basic Ratios

• sin θ = opposite / hypotenuse (MEMORISE THIS)
• cos θ = adjacent / hypotenuse (MEMORISE THIS)
• tan θ = opposite / adjacent (MEMORISE THIS)
• cosec θ = 1 / sin θ (MEMORISE THIS)
• sec θ = 1 / cos θ (MEMORISE THIS)
• cot θ = 1 / tan θ (MEMORISE THIS)

2. Pythagorean Identities

• sin²θ + cos²θ = 1 (MEMORISE THIS)
• 1 + tan²θ = sec²θ (MEMORISE THIS)
• 1 + cot²θ = cosec²θ (MEMORISE THIS)

3. Complementary Angle Identities

• sin(90° – θ) = cos θ (MEMORISE THIS)
• cos(90° – θ) = sin θ (MEMORISE THIS)
• tan(90° – θ) = cot θ (MEMORISE THIS)

4. Angle Sum & Difference Formulas (Given on exam sheet in most cases)

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Step-by-Step Method

Step 1: Identify the Given Information

• Look for:
• A right triangle with sides.
• An angle and a ratio (e.g., sin θ = 3/5).
• An identity to prove (e.g., "Prove that tan θ + cot θ = sec θ cosec θ").

Step 2: Draw a Right Triangle (If Needed)

• Label:
• Hypotenuse (longest side).
• Opposite side (across from the angle).
• Adjacent side (next to the angle).

Step 3: Use the Correct Formula

• If given a ratio (e.g., sin θ = 3/5), use Pythagorean theorem to find missing sides.
• If proving an identity, start with one side and simplify using known identities.

Step 4: Solve for Missing Values

• For ratios: Use sin²θ + cos²θ = 1 to find other ratios.
• For identities: Convert everything to sin and cos for easier simplification.

Step 5: Check for Quadrant Signs

• If θ is in Quadrant II, sin θ is positive, cos θ is negative.
• If θ is in Quadrant III, tan θ is positive, others negative.

Step 6: Verify Your Answer

• Plug in a known angle (e.g., θ = 30°) to check if your answer makes sense.

Worked Examples

Example 1 – Basic (Finding Missing Ratios)

Question: If sin θ = 3/5, find cos θ and tan θ.

Solution:
1. Draw a right triangle with: - Opposite = 3 - Hypotenuse = 5
2. Find adjacent side using Pythagorean theorem: - Adjacent = √(5² – 3²) = √(25 – 9) = √16 = 4
3. Find cos θ = adjacent / hypotenuse = 4/5
4. Find tan θ = opposite / adjacent = 3/4

Answer: cos θ = 4/5, tan θ = 3/4

What we did and why: - Used sin θ = opposite/hypotenuse to label sides. - Applied Pythagorean theorem to find the missing side. - Used cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent to find the other ratios.

Example 2 – Medium (Proving an Identity)

Question: Prove that tan θ + cot θ = sec θ cosec θ.

Solution:
1. Start with the left side (LHS): tan θ + cot θ
2. Convert to sin and cos: - tan θ = sin θ / cos θ - cot θ = cos θ / sin θ
3. Combine fractions: - LHS = (sin θ / cos θ) + (cos θ / sin θ) = (sin²θ + cos²θ) / (sin θ cos θ)
4. Use Pythagorean identity (sin²θ + cos²θ = 1): - LHS = 1 / (sin θ cos θ)
5. Rewrite right side (RHS): sec θ cosec θ - sec θ = 1 / cos θ - cosec θ = 1 / sin θ - RHS = (1 / cos θ)(1 / sin θ) = 1 / (sin θ cos θ)
6. LHS = RHS, so the identity is proven.

Answer: Proved.

What we did and why: - Converted all terms to sin and cos for easier simplification. - Used the Pythagorean identity to simplify the numerator. - Compared both sides to confirm the identity.

Example 3 – Exam-Style (Disguised Question)

Question: If sin θ + cos θ = √2, find sin θ cos θ.

Solution:
1. Square both sides: - (sin θ + cos θ)² = (√2)² - sin²θ + 2 sin θ cos θ + cos²θ = 2
2. Use Pythagorean identity (sin²θ + cos²θ = 1): - 1 + 2 sin θ cos θ = 2
3. Solve for sin θ cos θ: - 2 sin θ cos θ = 2 – 1 = 1 - sin θ cos θ = 1/2

Answer: 1/2

What we did and why: - Squared both sides to use the Pythagorean identity. - Simplified using sin²θ + cos²θ = 1. - Isolated the required term (sin θ cos θ).

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute checklist for trigonometry ratios and identities:

1. Memorise the 3 basic ratios (sin, cos, tan) and their reciprocals.
2. Know the Pythagorean identities by heart: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
3. For missing ratios, draw a triangle and use the Pythagorean theorem.
4. When proving identities, always start with one side and convert everything to sin and cos.
5. Watch for quadrant signs—sin is positive in Quadrants I and II, cos in I and IV.
6. If the question gives sin θ + cos θ, square it to use sin²θ + cos²θ = 1.

Now go solve those 5-mark questions in under a minute!