JEE Mathematics Trigonometry Heights and Distances

Trigonometry — Heights and Distances

What This Is and Why It Matters for JEE

Heights and Distances is a fundamental topic in Trigonometry, appearing in 2-3 questions every year. It's moderately difficult, with a slight emphasis on Advanced.

Prerequisites

• Trigonometric ratios (sine, cosine, tangent)
• Right-angled triangles
• Basic algebra (equations, formulas)

Quick Revision Path

• Review trigonometric ratios and right-angled triangles.
• Brush up on basic algebra.

Core Concepts (Exam-Focused)

• Height of a tower using tangent and opposite/adjacent sides.
• Distance between two points using cosine law and Pythagorean theorem.
• Shadow of an object using sine and angle of elevation.

Key Formulae

• tangent: tan(θ) = opposite / adjacent
• cosine law: c² = a² + b² - 2ab * cos(θ)
• Pythagorean theorem: c² = a² + b²

Step-by-Step Problem-Solving Strategy

1. Identify the given information (angles, sides, distances).
2. Determine the unknown quantity (height, distance, shadow).
3. Choose the relevant trigonometric concept (tangent, cosine, sine).
4. Set up the equation using the chosen concept.
5. Solve for the unknown quantity.
6. Check for multiple cases or special conditions (e.g., acute/obtuse angles).
7. Verify the dimensions of the answer (if applicable).

Important Graphs / Diagrams

• Right-angled triangles: Examiners test slope, area, and intercepts.
• Angle of elevation: Examiners test tangent, sine, and cosine relationships.

Typical JEE Question Patterns

• Find the minimum/maximum value of a trigonometric function.
• Compare time periods of two trigonometric functions.
• Find the distance between two points using the cosine law.

Common Mistakes & Exam Traps

• Mistake: Not checking for multiple cases or special conditions.
  Why it happens: Rushing or misreading the question.
  How to avoid it: Always check for multiple cases and special conditions.
  Exam board insight: Examiners penalize incorrect assumptions.
• Mistake: Not verifying dimensions of the answer.
  Why it happens: Lack of attention to detail.
  How to avoid it: Always verify dimensions of the answer.
  Exam board insight: Examiners deduct marks for incorrect dimensions.

Time-Saving Shortcuts

• Use the tangent ratio to find the height of a tower.
  Shortcut: tan(θ) = opposite / adjacent.
  Condition: θ is acute.

Practice MCQs (Exam-Style)

Question 1: Find the distance between two points A and B, given that the angle between the line joining A and B and the x-axis is 60°, and the coordinates of A and B are (3, 4) and (5, 6) respectively.

A) 10 B) 12 C) 15 D) 20

Answer: B) 12 Solution: Use the cosine law: c² = a² + b² - 2ab * cos(θ).
Common Wrong Answer: Option A, due to incorrect application of the cosine law.

Question 2: A tower stands on a horizontal plane. At a point 50 m away from the foot of the tower on the same plane, the angle of elevation of the top of the tower is 60°. Find the height of the tower.

A) 50 m B) 75 m C) 100 m D) 150 m

Answer: B) 75 m Solution: Use the tangent ratio: tan(θ) = opposite / adjacent.
Common Wrong Answer: Option A, due to incorrect application of the tangent ratio.

Question 3: (JEE Advanced level) A right-angled triangle has a hypotenuse of length 10 cm and one of its acute angles is 30°. Find the length of the side opposite the 30° angle.

A) 5 cm B) 7.5 cm C) 10 cm D) 15 cm

Answer: B) 7.5 cm Solution: Use the cosine law: c² = a² + b² - 2ab * cos(θ).
Common Wrong Answer: Option C, due to incorrect application of the cosine law.

Quick Revision Card (60-Second Summary)

• tangent: tan(θ) = opposite / adjacent
• cosine law: c² = a² + b² - 2ab * cos(θ)
• Pythagorean theorem: c² = a² + b²
• angle of elevation: tan(θ) = opposite / adjacent
• right-angled triangles: Examiners test slope, area, and intercepts.

If You Get Stuck in Exam

• Write down what you know (partial marks strategy).
• Eliminate distractors by checking units and dimensions.
• Skip and return if stuck, but check the question again before moving on.

Related JEE Topics

• Trigonometric identities: Used to simplify trigonometric expressions.
• Graphs of trigonometric functions: Used to visualize trigonometric relationships.
• Analytical geometry: Used to solve problems involving right-angled triangles.