Trigonometry — SSC CHSL Study Notes

Overview

Trigonometry in SSC CHSL tests your understanding of angular relationships in right triangles and their practical applications. This topic typically yields 2–3 questions in Tier 1, focusing heavily on standard identities, ratio calculations, and heights-and-distances word problems.

Mastery requires two things: instant recall of fundamental ratios (sin, cos, tan for standard angles) and fluent manipulation of Pythagorean and compound identities. Most questions are formula-driven with moderate calculation. Students who memorize the angle table (0°, 30°, 45°, 60°, 90°) and practice substitution in identities can score full marks here. The heights-and-distances subset — applying trigonometry to real-world scenarios involving towers, ladders, and shadows — is straightforward once you learn to draw the right triangle and identify the correct ratio.

Unlike geometry, trigonometry problems rarely involve deep reasoning. Speed and accuracy in applying the right identity or ratio is what differentiates top scorers. Practice enough to recognize patterns like "given tan θ, find sin θ + cos θ" or "shadow of a pole at angle 30°" instantly.

Key Concepts

• **Six trigonometric ratios** exist for any acute angle θ in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, cot θ = adjacent/opposite, sec θ = hypotenuse/adjacent, csc θ = hypotenuse/opposite. These definitions are the foundation of all problems.

• **Reciprocal identities** link pairs of ratios: sin θ = 1/csc θ, cos θ = 1/sec θ, tan θ = 1/cot θ. Also, tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. Use these to convert between ratios when needed.

• **Pythagorean identities** are the most tested: sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, 1 + cot² θ = csc² θ. These let you find one ratio when another is given.

• **Complementary angle identities** apply when angles add to 90°: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ. Useful for simplification in identities.

• **Standard angle values** (0°, 30°, 45°, 60°, 90°) must be memorized. For example, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3, sin 90° = 1. These appear in nearly every trigonometry question.

• **Heights and distances** problems model real-world scenarios using a right triangle where angle of elevation (looking up) or angle of depression (looking down) is given. Identify the height (opposite or adjacent), distance (base), and angle, then apply the correct ratio.

• **Multiple angle formulas** are rare in CHSL but know that sin 2θ = 2 sin θ cos θ and cos 2θ = cos² θ − sin² θ for occasional identity simplifications.

Formulas / Key Facts

**Basic Ratios:** sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent cot θ = 1/tan θ, sec θ = 1/cos θ, csc θ = 1/sin θ

**Pythagorean Identities:** sin² θ + cos² θ = 1 1 + tan² θ = sec² θ 1 + cot² θ = csc² θ

**Complementary Angles (sum = 90°):** sin(90° − θ) = cos θ, cos(90° − θ) = sin θ tan(90° − θ) = cot θ, cot(90° − θ) = tan θ sec(90° − θ) = csc θ, csc(90° − θ) = sec θ

**Standard Angle 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 | ∞ |

**Height-Distance Key:** tan θ = height / distance (when height is opposite to angle) Angle of elevation = angle above horizontal; angle of depression = angle below horizontal (measured from observer's level)

Worked Examples

**Example 1:** If sin θ = 3/5, find cos θ and tan θ (assuming θ is acute).

*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 (positive in first quadrant).

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 + tan² A) / (1 + cot² A).

*Solution:* Use identities: 1 + tan² A = sec² A and 1 + cot² A = csc² A. Expression becomes sec² A / csc² A. sec A = 1/cos A, csc A = 1/sin A, so sec² A / csc² A = (1/cos² A) / (1/sin² A) = sin² A / cos² A = tan² A.

**Answer:** tan² A.

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**Example 3:** A tower is 50 m tall. From a point on the ground, the angle of elevation to the top is 30°. Find the distance from the point to the base of the tower.

*Solution:* Draw right triangle: height (opposite) = 50 m, angle = 30°, distance (adjacent) = ? Use tan 30° = opposite / adjacent. 1/√3 = 50 / distance distance = 50√3 m ≈ 86.6 m.

**Answer:** 50√3 m.

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**Example 4:** If 3 tan θ = 4, find the value of (3 sin θ − 2 cos θ) / (3 sin θ + 2 cos θ).

*Solution:* tan θ = 4/3, so sin θ / cos θ = 4/3 → sin θ = 4k, cos θ = 3k (using Pythagoras: (4k)² + (3k)² = 1 → 25k² = 1 → k = 1/5). So sin θ = 4/5, cos θ = 3/5.

Substitute: Numerator = 3(4/5) − 2(3/5) = 12/5 − 6/5 = 6/5. Denominator = 3(4/5) + 2(3/5) = 12/5 + 6/5 = 18/5. Result = (6/5) / (18/5) = 6/18 = 1/3.

**Answer:** 1/3.

Common Mistakes

**Using degrees vs. ratios carelessly:** Students write "sin 30 = 1/2" without the degree symbol or confuse sin 30° (which is 1/2) with sin 30 radians (a different value). Always specify the angle unit; CHSL uses degrees exclusively. **Fix:** Memorize standard angle values in degrees and always include the ° symbol mentally.

**Forgetting the Pythagorean step:** Given tan θ = 3/4, students jump to sin θ = 3, cos θ = 4 without recognizing these must satisfy sin² + cos² = 1. **Fix:** Use opposite = 3k, adjacent = 4k, hypotenuse = 5k (by Pythagoras), then sin θ = 3/5, cos θ = 4/5.

**Mixing up angle of elevation and depression:** Treating both as the same or measuring from the wrong reference line (e.g., measuring angle of depression from the ground instead of the horizontal at observer's eye level). **Fix:** Angle of elevation = looking upward from horizontal; angle of depression = looking downward from horizontal at observer's height. Both form the same angle with the vertical in alternate interior angle scenarios.

**Incorrect identity substitution:** Writing 1 + tan² θ = csc² θ instead of sec² θ. **Fix:** Drill all three Pythagorean identities separately: sin²+cos²=1, 1+tan²=sec², 1+cot²=csc². Don't mix them.

**Sign errors in identities:** Assuming cos 2θ = 1 − 2sin² θ is the only form and forgetting cos 2θ = 2cos² θ − 1 or cos² θ − sin² θ (though these are rare in CHSL, the principle applies to substitution mistakes in simplification). **Fix:** Write out both sides of the identity step-by-step; don't skip algebraic steps.

Quick Reference

• **Memorize the angle table:** sin/cos/tan for 0°, 30°, 45°, 60°, 90° — non-negotiable for speed.
• **Three Pythagorean identities:** sin² + cos² = 1, 1 + tan² = sec², 1 + cot² = csc².
• **Complementary swap:** sin(90° − θ) = cos θ, tan(90° − θ) = cot θ.
• **Heights-distances formula:** tan(angle) = height / base distance — draw the triangle every time.
• **Reciprocal pairs:** sin ↔ csc, cos ↔ sec, tan ↔ cot.
• **Convert tan to sin/cos:** tan θ = sin θ / cos θ — useful when only tan is given and you need sin or cos separately.