Trigonometry — Study Notes for SSC MTS

Overview

Trigonometry in SSC MTS is limited but high-scoring. You'll face 1–2 questions testing trigonometric ratios (sin, cos, tan and their reciprocals) and a handful of fundamental identities. The exam does **not** expect height-distance problems or advanced angle transformations — focus entirely on ratio definitions, complementary angle relations, and the three Pythagorean identities.

Most questions are direct substitutions: given one ratio, find another; or simplify an expression using identities. A strong grasp of the six basic ratios for standard angles (0°, 30°, 45°, 60°, 90°) and comfort with algebraic manipulation of the three main identities will fetch you full marks. This topic connects to algebra (identities are equations) and mensuration (right triangles), so mastery here reinforces other chapters.

**Exam weight:** 1–2 questions out of 25 in Numerical Ability. Every mark counts in a competitive exam, and trigonometry questions are faster to solve than multi-step arithmetic once you memorize the essentials.

Key Concepts

• **Trigonometric ratios in a right triangle:** For an acute angle θ in a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The reciprocals are cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

• **Standard angles:** Memorize exact values for 0°, 30°, 45°, 60° and 90°. For example, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3. These appear in nearly every trigonometry question.

• **Pythagorean identities:** The three must-know identities are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ. These let you convert one ratio into another or simplify complex expressions.

• **Complementary angle relations:** sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, and their reciprocals follow the same swap. Use these when an angle like 60° and 30° appear together (since 60° + 30° = 90°).

• **Reciprocal relationships:** cosec θ and sin θ are reciprocals, as are sec θ and cos θ, and cot θ and tan θ. If you know one, the other is immediate: sec θ = 1/cos θ, so if cos θ = 3/5 then sec θ = 5/3.

• **Ratio interconversions:** tan θ = sin θ/cos θ and cot θ = cos θ/sin θ. These are handy when simplifying fractions involving multiple ratios.

Formulas / Key Facts

**Trigonometric Ratios (right triangle with angle θ):**

• sin θ = perpendicular / hypotenuse
• cos θ = base / hypotenuse
• tan θ = perpendicular / base
• cosec θ = hypotenuse / perpendicular = 1/sin θ
• sec θ = hypotenuse / base = 1/cos θ
• cot θ = base / perpendicular = 1/tan θ

**Standard Angle Values (memorize this table):** | Angle | 0° | 30° | 45° | 60° | 90° | |-------|-----|------|-------|------|------| | sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | | cos | 1 | √3/2| 1/√2 | 1/2 | 0 | | tan | 0 | 1/√3| 1 | √3 | undefined |

**Pythagorean Identities:**

• sin² θ + cos² θ = 1
• 1 + tan² θ = sec² θ
• 1 + cot² θ = cosec² θ

**Complementary Angle Relations:**

• 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

**Ratio Relations:**

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

Worked Examples

**Example 1:** If sin θ = 3/5, find cos θ and tan θ.

*Solution:* Use sin² θ + cos² θ = 1. (3/5)² + cos² θ = 1 9/25 + cos² θ = 1 cos² θ = 1 − 9/25 = 16/25 cos θ = 4/5 (taking positive value for acute angle).

Now tan θ = sin θ / cos θ = (3/5) / (4/5) = 3/4.

**Answer:** cos θ = 4/5, tan θ = 3/4.

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**Example 2:** Simplify: (1 − sin² θ) sec² θ.

*Solution:* Recognize 1 − sin² θ = cos² θ (from the first Pythagorean identity). So the expression becomes cos² θ · sec² θ. Since sec θ = 1/cos θ, we have sec² θ = 1/cos² θ. Thus cos² θ · (1/cos² θ) = 1.

**Answer:** 1.

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**Example 3:** If tan 30° = 1/√3, find cot 60°.

*Solution:* Use the complementary angle relation: tan(90° − θ) = cot θ. Here 30° + 60° = 90°, so tan 30° = cot 60°. Therefore cot 60° = 1/√3.

**Answer:** 1/√3.

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**Example 4:** Evaluate: sin² 45° + cos² 45°.

*Solution:* We know sin 45° = 1/√2 and cos 45° = 1/√2. sin² 45° = (1/√2)² = 1/2 cos² 45° = (1/√2)² = 1/2 Sum = 1/2 + 1/2 = 1.

Alternatively, use the identity sin² θ + cos² θ = 1 directly — true for any angle.

**Answer:** 1.

Common Mistakes

• **Forgetting that cosec, sec, cot are reciprocals, not complements:** Students often confuse cosec θ with "complement of sec." Remember: cosec θ = 1/sin θ, not a complement. The complementary swap is sin(90° − θ) = cos θ.

• **Misapplying the Pythagorean identity to tan and cot:** The identity is 1 + tan² θ = sec² θ, *not* tan² θ + cot² θ = 1. Each Pythagorean identity has a specific form — don't mix them.

• **Sign errors with square roots:** When you solve cos² θ = 16/25, you get cos θ = ±4/5. For acute angles in SSC MTS (the typical context), all six ratios are positive, so pick the positive root. Don't drop the sign consideration.

• **Not memorizing standard angles:** Trying to derive sin 30° = 1/2 in the exam wastes time. Commit the 0°–30°–45°–60°–90° table to memory. Slow recall here kills your speed advantage.

• **Confusing tan θ = sin θ/cos θ with tan θ = cos θ/sin θ:** The correct relation is tan θ = sin θ / cos θ. The reciprocal is cot θ = cos θ / sin θ. Mixing these up flips your answer.

Quick Reference

• **Six ratios:** sin, cos, tan (primary); cosec = 1/sin, sec = 1/cos, cot = 1/tan (reciprocals).

• **Core identity:** sin² θ + cos² θ = 1 — use this to find one ratio when the other is given.

• **Tan/sec identity:** 1 + tan² θ = sec² θ — helpful when tan or sec appears alone.

• **Standard values:** sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2; cos is the reverse sequence; tan 45° = 1.

• **Complementary swap:** sin(90° − θ) = cos θ; tan(90° − θ) = cot θ — use when you see angles like 30° and 60° together.

• **Practice speed:** Trigonometry questions are quick wins if you know identities cold — aim to solve each in under 60 seconds.