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Quantitative Aptitude

Trigonometric Ratios & Identities

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Concepts (3)

Master trigonometric ratios (sin, cos, tan, cosec, sec, cot) and standard angle values (0°, 30°, 45°, 60°, 90°) for quick problem-solving. Essential for SSC CGL.

Core Formulas

Trigonometric ratios relate the angles of a right-angled triangle to the ratio of its sides. For an acute angle θ in a right triangle:

  • Sine (sin θ) = Opposite side / Hypotenuse
  • Cosine (cos θ) = Adjacent side / Hypotenuse
  • Tangent (tan θ) = Opposite side / Adjacent side

Reciprocal Ratios:

  • Cosecant (cosec θ) = 1 / sin θ = Hypotenuse / Opposite side
  • Secant (sec θ) = 1 / cos θ = Hypotenuse / Adjacent side
  • Cotangent (cot θ) = 1 / tan θ = Adjacent side / Opposite side

Standard Angle Values Table:

Angle (θ)0°30°45°60°90°
sin θ01/21/√2√3/21
cos θ1√3/21/√21/20
tan θ01/√31√3Undefined

Worked Example 1

Q: In a right triangle with sides 3, 4, and 5, if the angle opposite to side 3 is θ, what is sin θ?

Solution:

  1. Identify the sides: Hypotenuse = 5 (longest side). Opposite side to θ = 3. Adjacent side to θ = 4.
  2. Apply the formula for sin θ: sin θ = Opposite / Hypotenuse
  3. Substitute values: sin θ = 3 / 5

Answer: sin θ = 3/5

Worked Example 2

Q: Evaluate: sin 30° + cos 60° - tan 45°

Solution:

  1. Recall standard values from the table:
    • sin 30° = 1/2
    • cos 60° = 1/2
    • tan 45° = 1
  2. Substitute these values into the expression: 1/2 + 1/2 - 1
  3. Perform the arithmetic: 1 - 1 = 0

Answer: 0

Shortcuts & Tricks

  1. SOH CAH TOA: Mnemonic for basic ratios:
    • Sin = Opposite / Hypotenuse
    • Cos = Adjacent / Hypotenuse
    • Tan = Opposite / Adjacent
  2. Hand Trick for sin/cos: For 0°, 30°, 45°, 60°, 90°:
    • Hold your left hand up. Thumb is 0°, index 30°, middle 45°, ring 60°, pinky 90°.
    • For sin: Count fingers below the chosen finger, take square root, divide by 2. (e.g., sin 30°: 1 finger below index, so √1/2 = 1/2)
    • For cos: Count fingers above the chosen finger, take square root, divide by 2. (e.g., cos 30°: 3 fingers above index, so √3/2)
  3. Complementary Angles: For 0° < θ < 90°:
    • sin (90° - θ) = cos θ
    • cos (90° - θ) = sin θ
    • tan (90° - θ) = cot θ
    • cot (90° - θ) = tan θ
    • sec (90° - θ) = cosec θ
    • cosec (90° - θ) = sec θ
    • Quick application: If sin 20° = cos x, then x = 70° (90-20).

Common Mistakes

  1. Confusing Opposite/Adjacent: Always identify the angle first, then the side opposite to it and the side adjacent to it. The hypotenuse is always opposite the 90° angle.
  2. Incorrect Standard Values: Misremembering values like sin 60° as 1/2 instead of √3/2. Memorize the table thoroughly or use the hand trick.
  3. Reciprocal Errors: Confusing cosec with cos, or sec with sin. Remember cosec goes with sin and sec with cos (the 'co' in cos/cot/cosec helps link them, but be careful).
  4. Undefined Values: Forgetting that tan 90° and cot 0° are undefined.

Derivation (brief)

Trigonometric ratios are fundamentally derived from the properties of similar right-angled triangles. For any given acute angle, the ratio of any two sides in a right triangle remains constant, regardless of the triangle's size. This is because all right triangles with the same acute angle are similar. For instance, sin θ is the ratio of the side opposite to angle θ to the hypotenuse. As long as θ is the same, this ratio will be the same for all such triangles.

Advanced Examples

Q: If tan A = 1 and sin B = √3/2, find the value of (cos A + sin B) / (sec A - cot B).

Solution:

  1. From tan A = 1, we know A = 45°.
    • cos A = cos 45° = 1/√2
    • sec A = sec 45° = 1 / cos 45° = √2
  2. From sin B = √3/2, we know B = 60°.
    • cot B = cot 60° = 1 / tan 60° = 1/√3
  3. Substitute these values into the expression: (1/√2 + √3/2) / (√2 - 1/√3)
  4. Simplify the numerator and denominator: Numerator: (√3 + 1) / (2√2) Denominator: (√6 - 1) / √3
  5. Divide the numerator by the denominator: [(√3 + 1) / (2√2)] * [√3 / (√6 - 1)] = (3 + √3) / (2√2(√6 - 1)) = (3 + √3) / (2√12 - 2√2) = (3 + √3) / (4√3 - 2√2)
  6. Rationalize the denominator (multiply by conjugate): = [(3 + √3) / (4√3 - 2√2)] * [(4√3 + 2√2) / (4√3 + 2√2)] = (12√3 + 6√2 + 12 + 2√6) / ( (4√3)² - (2√2)² ) = (12√3 + 6√2 + 12 + 2√6) / (48 - 8) = (12 + 12√3 + 6√2 + 2√6) / 40 = (6 + 6√3 + 3√2 + √6) / 20

Variation Types

  1. Direct Evaluation: Simple expressions involving standard angles (e.g., Ex 2).
  2. Triangle-based: Given sides of a right triangle, find ratios (e.g., Ex 1).
  3. Equation Solving: Find an unknown angle or side given a trigonometric equation (e.g., If sin θ = cos 2θ, find θ). Use complementary angle relations.
  4. Identity Application: Questions that combine standard angles with basic trigonometric identities (e.g., sin²θ + cos²θ = 1).
  5. Conditional Questions: Given one ratio, find others (e.g., If sin θ = 3/5, find tan θ). Draw a right triangle to find the third side.

Time-Saving Methods

  1. Absolute Memorization: The standard angle table should be etched in your memory. Practice recalling values instantly.
  2. Recognize Complementary Pairs: When you see sin 20° and cos 70° in an expression, immediately know they are equal. This simplifies many problems.
  3. Pythagorean Triplets: Memorize common triplets (3,4,5; 5,12,13; 7,24,25; 8,15,17) to quickly find the third side of a right triangle when two are given, saving calculation time.
  4. Mental Math: Practice adding, subtracting, multiplying, and dividing fractions and square roots quickly without writing down every step.
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Master Pythagorean identities (sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=cosec²θ) for quick simplification and problem-solving in SSC CGL.

Trigonometric identities are fundamental equations true for all values of the variables for which the expressions are defined. Mastering these identities is crucial for speed and accuracy in SSC CGL exams, as they simplify complex expressions and solve equations efficiently.

Core Formulas

  1. Pythagorean Identities:
    • sin²θ + cos²θ = 1
    • 1 + tan²θ = sec²θ (or sec²θ - tan²θ = 1)
    • 1 + cot²θ = cosec²θ (or cosec²θ - cot²θ = 1)
  2. Quotient Identities:
    • tan θ = sin θ / cos θ
    • cot θ = cos θ / sin θ
  3. Reciprocal Identities:
    • cosec θ = 1 / sin θ
    • sec θ = 1 / cos θ
    • cot θ = 1 / tan θ

Worked Example 1

Question: If sin θ + cos θ = √2, then the value of sin θ × cos θ is:

Solution:

  1. Given sin θ + cos θ = √2.
  2. Square both sides: (sin θ + cos θ)² = (√2)²
  3. Expand: sin²θ + cos²θ + 2sin θ cos θ = 2
  4. Apply the identity sin²θ + cos²θ = 1: 1 + 2sin θ cos θ = 2
  5. Subtract 1 from both sides: 2sin θ cos θ = 1
  6. Divide by 2: sin θ cos θ = 1/2

Worked Example 2

Question: The value of (1 + tan²A)(1 + sin A)(1 - sin A) is:

Solution:

  1. Identify the first part: (1 + tan²A). Using the identity 1 + tan²θ = sec²θ, this becomes sec²A.
  2. Identify the second part: (1 + sin A)(1 - sin A). This is in the form (a+b)(a-b) = a² - b², so it becomes 1² - sin²A = 1 - sin²A.
  3. Using the identity sin²θ + cos²θ = 1, we know 1 - sin²A = cos²A.
  4. Substitute these simplified forms back into the original expression: sec²A × cos²A
  5. Recall that sec A = 1 / cos A. So, sec²A = 1 / cos²A.
  6. Substitute this: (1 / cos²A) × cos²A = 1

Shortcuts & Tricks

  • Conjugate Pairs: If secθ + tanθ = k, then secθ - tanθ = 1/k. Similarly, if cosecθ + cotθ = k, then cosecθ - cotθ = 1/k. This saves immense time in problems involving these sums/differences.
  • Value Substitution: For problems without specific angles, if the options are numerical, try substituting a common angle like θ = 45° (where sin, cos, tan, cot, sec, cosec are all defined and easy to calculate). This can quickly eliminate options or lead to the answer.
  • Convert to Sin/Cos: When stuck with complex expressions involving tan, cot, sec, cosec, convert everything to sin and cos. This often reveals simpler forms.

Common Mistakes

  • Sign Errors: Confusing sec²θ - tan²θ = 1 with tan²θ - sec²θ = 1 (which is -1). Same for cosec²θ - cot²θ = 1.
  • Algebraic Errors: Incorrectly expanding (a+b)² or (a-b)², or making mistakes with fractions when converting to sin/cos.
  • Forgetting sin²θ + cos²θ = 1: This is the most basic identity, yet students sometimes overlook its application in complex problems.

Derivation (brief)

Trigonometric identities are derived from the Pythagorean theorem applied to a right-angled triangle. Consider a right-angled triangle with angle θ, opposite side y, adjacent side x, and hypotenuse r.

By the Pythagorean theorem: x² + y² = r²

  1. Deriving sin²θ + cos²θ = 1: Divide the entire equation x² + y² = r² by r²: (x²/r²) + (y²/r²) = (r²/r²) (x/r)² + (y/r)² = 1 Since cos θ = x/r and sin θ = y/r, this becomes cos²θ + sin²θ = 1.

  2. Deriving 1 + tan²θ = sec²θ: Divide the entire equation x² + y² = r² by x²: (x²/x²) + (y²/x²) = (r²/x²) 1 + (y/x)² = (r/x)² Since tan θ = y/x and sec θ = r/x, this becomes 1 + tan²θ = sec²θ.

  3. Deriving 1 + cot²θ = cosec²θ: Divide the entire equation x² + y² = r² by y²: (x²/y²) + (y²/y²) = (r²/y²) (x/y)² + 1 = (r/y)² Since cot θ = x/y and cosec θ = r/y, this becomes cot²θ + 1 = cosec²θ.

Advanced Examples

Question: If secθ + tanθ = P, find sinθ in terms of P.

Solution:

  1. We know sec²θ - tan²θ = 1.
  2. Factor the left side: (secθ - tanθ)(secθ + tanθ) = 1.
  3. Substitute the given secθ + tanθ = P: (secθ - tanθ)P = 1 So, secθ - tanθ = 1/P (Equation 1)
  4. We have secθ + tanθ = P (Equation 2)
  5. Add Equation 1 and Equation 2: (secθ - tanθ) + (secθ + tanθ) = 1/P + P 2secθ = (1 + P²) / P secθ = (P² + 1) / (2P)
  6. Since cosθ = 1/secθ: cosθ = 2P / (P² + 1)
  7. Now, to find sinθ, use sin²θ = 1 - cos²θ: sin²θ = 1 - [2P / (P² + 1)]² sin²θ = 1 - [4P² / (P² + 1)²] sin²θ = [(P² + 1)² - 4P²] / (P² + 1)² sin²θ = [P⁴ + 2P² + 1 - 4P²] / (P² + 1)² sin²θ = [P⁴ - 2P² + 1] / (P² + 1)² sin²θ = (P² - 1)² / (P² + 1)²
  8. Taking the square root (assuming sinθ is positive for acute angles): sinθ = (P² - 1) / (P² + 1)

Variation Types

  • Direct Identity Application: Simple problems requiring direct use of one or two identities.
  • Simplification of Complex Expressions: Questions asking to simplify a large trigonometric expression to a single value or term.
  • Conditional Problems: Given a relationship (e.g., sinθ + cosθ = k), find another related expression (e.g., sinθ - cosθ or sin³θ + cos³θ).
  • Problems with Specific Angles: While identities are general, questions often combine them with specific angle values (e.g., sin 30°, tan 45°).
  • Proof-based (indirectly): Questions asking which option is equivalent to a given expression, essentially asking you to prove an identity.

Time-Saving Methods

  • Memorize secθ ± tanθ and cosecθ ± cotθ relations: Knowing that secθ - tanθ = 1/(secθ + tanθ) is a huge time-saver. Practice deriving sinθ and cosθ from secθ + tanθ = P so you can recall the results quickly.
  • Recognize a² - b² patterns: (1 - sin²θ) is cos²θ, (sec²θ - 1) is tan²θ, etc. These are frequent.
  • Smart Angle Substitution: If an expression is true for all θ (and options are numerical), θ = 45° is often the easiest to calculate. Be careful with angles that make denominators zero (e.g., tan 90° is undefined).
  • Practice, Practice, Practice: The more you solve, the faster you'll recognize patterns and apply the correct identity without hesitation.
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Master simplifying trigonometric expressions using fundamental identities and algebraic manipulation. Prioritize quick recall and smart substitutions for speed and accuracy in SSC CGL.

Core Formula

Simplification of trigonometric expressions heavily relies on fundamental identities and algebraic manipulation. Key identities to remember:

  1. Pythagorean Identities:
    • sin²θ + cos²θ = 1
    • sec²θ - tan²θ = 1
    • cosec²θ - cot²θ = 1
  2. Reciprocal Identities:
    • sin θ = 1/cosec θ, cos θ = 1/sec θ, tan θ = 1/cot θ
  3. Quotient Identities:
    • tan θ = sin θ / cos θ, cot θ = cos θ / sin θ
  4. Algebraic Identities: Often, trigonometric expressions simplify using (a+b)², (a-b)², a²-b², (a+b)³, etc.
    • A very common one: (sin θ + cos θ)² = sin²θ + cos²θ + 2 sin θ cos θ = 1 + 2 sin θ cos θ
    • Similarly: (sin θ - cos θ)² = 1 - 2 sin θ cos θ

Worked Example 1

Question: If sin θ + cos θ = √2, find the value of sin θ cos θ. Solution: Given sin θ + cos θ = √2 Square both sides: (sin θ + cos θ)² = (√2)² sin²θ + cos²θ + 2 sin θ cos θ = 2 Using the identity sin²θ + cos²θ = 1: 1 + 2 sin θ cos θ = 2 2 sin θ cos θ = 2 - 1 2 sin θ cos θ = 1 sin θ cos θ = 1/2

Worked Example 2

Question: Simplify (1 + tan²θ) / (1 + cot²θ). Solution: We know the Pythagorean identities: sec²θ = 1 + tan²θ cosec²θ = 1 + cot²θ Substitute these into the expression: (1 + tan²θ) / (1 + cot²θ) = sec²θ / cosec²θ Now, express sec θ and cosec θ in terms of sin θ and cos θ: sec θ = 1/cos θ cosec θ = 1/sin θ So, sec²θ / cosec²θ = (1/cos²θ) / (1/sin²θ) = (1/cos²θ) * (sin²θ/1) = sin²θ / cos²θ = tan²θ

Shortcuts & Tricks

  1. Angle Substitution (Value Putting): If the expression is independent of the angle (i.e., the final answer is a constant or a simple trig ratio without a specific angle), try substituting a convenient angle like θ = 45°. This often simplifies calculations significantly. Be careful not to use angles that make denominators zero (e.g., θ=0° for cot θ, θ=90° for tan θ).
  2. Convert to Sin/Cos: When stuck, converting all terms to sin θ and cos θ is a reliable strategy. This often reveals common denominators or Pythagorean identities.
  3. Recognize Algebraic Patterns: Look for (a+b)², (a-b)², a²-b², (a+b)³, etc. For example, (sec θ + tan θ)(sec θ - tan θ) immediately simplifies to sec²θ - tan²θ = 1.

Common Mistakes

  1. Algebraic Errors: Incorrect expansion of (a+b)² or (a-b)², or errors in combining fractions. Always double-check your basic algebra.
  2. Incorrect Identity Application: Misremembering or misapplying identities (e.g., confusing sec²θ - tan²θ = 1 with tan²θ - sec²θ = 1).
  3. Division by Zero: Using angle substitution where the chosen angle makes a term undefined (e.g., tan 90° or cot 0°). Always check for domain restrictions.

Derivation (brief)

1. sin²θ + cos²θ = 1: Consider a right-angled triangle with angle θ. Let the perpendicular be p, base b, and hypotenuse h. By Pythagoras theorem, p² + b² = h². We know sin θ = p/h and cos θ = b/h. So, sin²θ + cos²θ = (p/h)² + (b/h)² = p²/h² + b²/h² = (p² + b²) / h² = h²/h² = 1. 2. sec²θ - tan²θ = 1: Divide sin²θ + cos²θ = 1 by cos²θ (assuming cos θ ≠ 0): (sin²θ / cos²θ) + (cos²θ / cos²θ) = 1 / cos²θ tan²θ + 1 = sec²θ Rearranging gives sec²θ - tan²θ = 1.

Advanced Examples

Question 1: If x = a sin θ - b cos θ and y = a cos θ + b sin θ, find x² + y². Solution: x² = (a sin θ - b cos θ)² = a² sin²θ + b² cos²θ - 2ab sin θ cos θ y² = (a cos θ + b sin θ)² = a² cos²θ + b² sin²θ + 2ab sin θ cos θ Adding x² and y²: x² + y² = (a² sin²θ + b² cos²θ - 2ab sin θ cos θ) + (a² cos²θ + b² sin²θ + 2ab sin θ cos θ) x² + y² = a² sin²θ + b² cos²θ + a² cos²θ + b² sin²θ (The 2ab sin θ cos θ terms cancel out) x² + y² = a²(sin²θ + cos²θ) + b²(cos²θ + sin²θ) Using sin²θ + cos²θ = 1: x² + y² = a²(1) + b²(1) x² + y² = a² + b²

Question 2: If tan θ + cot θ = 2, find tan⁵θ + cot⁵θ. Solution: Given tan θ + cot θ = 2 We know cot θ = 1/tan θ. So, tan θ + 1/tan θ = 2 Multiply by tan θ: tan²θ + 1 = 2 tan θ tan²θ - 2 tan θ + 1 = 0 (tan θ - 1)² = 0 This implies tan θ = 1. If tan θ = 1, then cot θ = 1/tan θ = 1/1 = 1. Therefore, tan⁵θ + cot⁵θ = (1)⁵ + (1)⁵ = 1 + 1 = 2.

Variation Types

  1. Expressions with Higher Powers: Problems like sin⁴θ + cos⁴θ (which can be written as (sin²θ + cos²θ)² - 2 sin²θ cos²θ = 1 - 2 sin²θ cos²θ) or sin⁶θ + cos⁶θ.
  2. Conditional Problems: Given a relation (e.g., sin θ + cos θ = k), find another related expression (e.g., sin θ - cos θ or sec θ + cosec θ). These often involve squaring or algebraic manipulation.
  3. Expressions to a Constant Value: Simplifying complex expressions to a single numerical value, often using angle substitution as a shortcut if the result is independent of the angle.

Time-Saving Methods

  1. Smart Angle Substitution: For expressions that simplify to a constant, θ = 45° is often the quickest as sin 45° = cos 45° = 1/√2, tan 45° = cot 45° = 1, sec 45° = cosec 45° = √2. If tan or cot terms are involved, θ = 30° or 60° can also be useful. Avoid 0° or 90° if tan, cot, sec, cosec are present, unless you're sure they won't lead to undefined terms.
  2. Recognizing (a+b)² + (a-b)² or (a+b)² - (a-b)²: These simplify to 2(a²+b²) and 4ab respectively. For example, (sin θ + cos θ)² + (sin θ - cos θ)² = 2(sin²θ + cos²θ) = 2(1) = 2.
  3. Memorize Common Results: For instance, if tan θ + cot θ = 2, then tan θ = 1 and cot θ = 1. This is a frequently appearing pattern.
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Start Lesson: Trigonometric Ratios & Standard Angles