Trigonometry — Study Notes

Overview

Trigonometry appears regularly in Railway Group D exams, typically 2–4 questions per paper. The syllabus is limited to basics: trigonometric ratios of standard angles, fundamental identities, and simple height-distance problems. Unlike advanced math courses, you will not encounter inverse functions, multiple-angle formulas, or complex proofs.

Mastery here means two things: instant recall of ratios for 0°, 30°, 45°, 60°, and 90°, and fluent use of Pythagorean identities. Height-and-distance problems test whether you can draw a right triangle from a word problem, label it correctly, and apply the appropriate ratio. Most errors come from mixing up opposite and adjacent sides or using the wrong angle. Build accuracy through repetition; speed will follow.

Trigonometry overlaps with geometry (especially right triangles) and mensuration (calculating inaccessible heights). A strong grasp here improves your performance across multiple Mathematics topics.

Key Concepts

• **Trigonometric ratios** are defined for acute angles in a right triangle. For angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, and the reciprocals cot θ = 1/tan θ, sec θ = 1/cos θ, csc θ = 1/sin θ.

• **Standard angles** (0°, 30°, 45°, 60°, 90°) have exact ratio values that must be memorized. These appear in nearly every trigonometry question.

• **Pythagorean identities** connect sin, cos, and tan: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ. These are tools for simplification and substitution.

• **Complementary angle relations**: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ. Use these when the problem mixes angles like 30° and 60°.

• **Height and distance problems** translate real-world scenarios (towers, ladders, kites) into right triangles. The "angle of elevation" is measured upward from horizontal; "angle of depression" is measured downward from horizontal.

• Ratios and identities must be automatic. Hesitation on sin 30° = 1/2 costs time on a timed exam. Drill the table until it is second nature.

Formulas / Key Facts

**Trigonometric Ratios for Standard Angles (Memorize This Table):**

| Angle | 0° | 30° | 45° | 60° | 90° | |---------|-----|-----------|-----------|-----------|-----------| | sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 | | cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 | | tan θ | 0 | 1/√3 | 1 | √3 | undefined | | cot θ | ∞ | √3 | 1 | 1/√3 | 0 | | sec θ | 1 | 2/√3 | √2 | 2 | undefined | | csc θ | ∞ | 2 | √2 | 2/√3 | 1 |

**Fundamental Identities:**

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

**Complementary Angle Formulas:**

• 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

**Reciprocal Relations:**

• csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ
• tan θ = sin θ / cos θ; cot θ = cos θ / sin θ

Worked Examples

**Example 1: Simplify 3 sin²30° + 4 cos²60° − 2 tan²45°**

Step 1: Substitute values from the table. sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1.

Step 2: Compute each term. 3(1/2)² = 3 × 1/4 = 3/4 4(1/2)² = 4 × 1/4 = 1 2(1)² = 2

Step 3: Combine. 3/4 + 1 − 2 = 3/4 − 1 = 3/4 − 4/4 = −1/4.

**Answer: −1/4**

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**Example 2: If sin θ = 3/5, find cos θ and tan θ.**

Step 1: Use the identity sin²θ + cos²θ = 1. (3/5)² + cos²θ = 1 9/25 + cos²θ = 1 cos²θ = 1 − 9/25 = 16/25 cos θ = 4/5 (take positive value for acute angle)

Step 2: Find tan θ. 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 3: A ladder 10 m long leans against a wall, making an angle of 60° with the ground. How high does it reach on the wall?**

Step 1: Draw a right triangle. Ladder = hypotenuse = 10 m, angle with ground = 60°, height on wall = opposite side.

Step 2: Use sin 60° = opposite / hypotenuse. sin 60° = height / 10 √3/2 = height / 10 height = 10 × √3/2 = 5√3 m ≈ 8.66 m.

**Answer: 5√3 m or approximately 8.66 m**

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**Example 4: From the top of a 30 m tower, the angle of depression to a point on the ground is 30°. Find the distance of the point from the base of the tower.**

Step 1: Angle of depression from top = angle of elevation from bottom = 30° (alternate angles).

Step 2: Height = 30 m (opposite side), distance = adjacent side. tan 30° = opposite / adjacent 1/√3 = 30 / distance distance = 30√3 m ≈ 51.96 m.

**Answer: 30√3 m or approximately 52 m**

Common Mistakes

• **Mixing up sin and cos for complementary angles**: Students write sin 60° = 1/2 (wrong; sin 60° = √3/2). Always double-check the table.

• **Forgetting to simplify surds**: Leaving tan 30° as 1/√3 instead of rationalizing to √3/3. While both are correct, exams often expect rationalized denominators.

• **Confusing opposite and adjacent**: In tan θ = opposite/adjacent, "opposite" is opposite **the angle θ**, not opposite the right angle. Draw and label the triangle clearly.

• **Using degrees instead of the exact value**: Writing sin 45° = 0.707 instead of 1/√2. Exact form is safer and often required for further algebra.

• **Angle of elevation vs angle of depression**: Both are measured from the horizontal, not the vertical. Students sometimes measure from the wrong baseline, flipping the triangle setup.

Quick Reference

• **sin²θ + cos²θ = 1** — the most-used identity; memorize it.
• **sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3** — half the standard-angle questions use 30°.
• **sin 45° = cos 45° = 1/√2, tan 45° = 1** — the symmetric angle.
• **sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3** — complements 30°.
• **Angle of elevation/depression = horizontal reference** — draw the triangle from the observer's eye level.
• **For heights: use sin or tan; for distances: use cos or tan** — pick the ratio that connects the known and unknown sides.