Mathematics Class 10 Applications of Trigonometry Heights and Distances

CHAPTER: APPLICATIONS OF TRIGONOMETRY: HEIGHTS AND DISTANCES

1. PREREQUISITES

Before starting this chapter, students should already know the following concepts:

• TRIGONOMETRY FUNDAMENTALS: Students should be familiar with the basic trigonometric ratios (sine, cosine, and tangent), and their reciprocal ratios (cosecant, secant, and cotangent).
• RIGHT TRIANGLE TRIGONOMETRY: Students should know how to use trigonometry to solve problems involving right triangles, including finding missing sides and angles.
• ANGLES AND THEIR MEASUREMENTS: Students should be able to measure and convert between different units of angles, such as degrees, radians, and grades.

2. MASTER ORGANIZER

3. FORMULAS & THEOREMS

4. DIAGRAMS TO KNOW

5. RAPID REVISION SHEET

• The sine, cosine, and tangent ratios are used to find missing sides and angles in a right triangle.
• The sine rule is used to find missing sides in a triangle.
• The cosine rule is used to find missing sides in a triangle.
• The area of a triangle can be found using the formula Area = 0.5 * base * height.
• The angle of elevation or depression can be found using the tangent ratio.
• When using the sine, cosine, or tangent ratios, make sure to use the correct ratio for the given angle.
• When using the sine rule, make sure to use the correct ratio for the given angle.
• When using the cosine rule, make sure to square the correct side.
• When finding the area of a triangle, make sure to use the correct formula for the given type of triangle.
• When finding the angle of elevation or depression, make sure to use the correct trigonometric ratio.

6. STEP‑BY‑STEP PROBLEM SOLVER

Problem Type 1: Finding a Missing Side in a Right Triangle

Problem: In a right triangle, the length of the hypotenuse is 10 cm, and the length of the opposite side is 6 cm. Find the length of the adjacent side.

Solution:

1 → Use the sine ratio to find the length of the adjacent side: sin(A) = opposite side / hypotenuse 2 → Rearrange the formula to solve for the adjacent side: adjacent side = hypotenuse * sin(A) 3 → Plug in the values: adjacent side = 10 * sin(A) 4 → Use the sine ratio to find the value of sin(A): sin(A) = 6 / 10 5 → Simplify the expression: sin(A) = 0.6 6 → Plug in the value of sin(A) into the expression for the adjacent side: adjacent side = 10 * 0.6 7 → Simplify the expression: adjacent side = 6 cm

Common mistakes to avoid:

• Using the wrong trigonometric ratio for the given angle
• Forgetting to use the correct ratio for the given angle

Problem Type 2: Finding a Missing Angle in a Right Triangle

Problem: In a right triangle, the length of the hypotenuse is 10 cm, and the length of the opposite side is 6 cm. Find the measure of the angle opposite the side of length 6 cm.

Solution:

1 → Use the sine ratio to find the measure of the angle: sin(A) = opposite side / hypotenuse 2 → Rearrange the formula to solve for the angle: A = arcsin(opposite side / hypotenuse) 3 → Plug in the values: A = arcsin(6 / 10) 4 → Simplify the expression: A = arcsin(0.6) 5 → Use a calculator to find the value of A: A ≈ 36.87°

Common mistakes to avoid:

• Using the wrong trigonometric ratio for the given angle
• Forgetting to use the correct ratio for the given angle

Problem Type 3: Finding the Height of a Building

Problem: A building is 50 m tall, and the angle of elevation from the ground to the top of the building is 30°. Find the distance from the base of the building to the point directly below the top of the building.

Solution:

1 → Use the tangent ratio to find the distance: tan(A) = opposite side / adjacent side 2 → Rearrange the formula to solve for the adjacent side: adjacent side = opposite side / tan(A) 3 → Plug in the values: adjacent side = 50 / tan(30°) 4 → Simplify the expression: adjacent side = 50 / 0.57735 5 → Simplify the expression: adjacent side ≈ 86.60 m

Common mistakes to avoid:

• Using the wrong trigonometric ratio for the given angle
• Forgetting to use the correct ratio for the given angle

7. COMMON CONFUSIONS SHEET

• Sine Rule vs Cosine Rule → The sine rule is used to find missing sides in a triangle, while the cosine rule is used to find missing sides in a triangle.
• Angle of Elevation vs Angle of Depression → The angle of elevation is the angle between the line of sight and the horizontal, while the angle of depression is the angle between the line of sight and the horizontal.
• Sine, Cosine, and Tangent Graphs → The sine, cosine, and tangent graphs show the values of sine, cosine, and tangent for different angles.

8. COMMON MISTAKES & TRAPS

• Mistake/Trap: Using the wrong trigonometric ratio for the given angle.
  • Why it happens: Students may use the wrong ratio because they are not paying attention to the given angle.
  • How to avoid: Make sure to use the correct ratio for the given angle.
• Mistake/Trap: Forgetting to use the correct ratio for the given angle.
  • Why it happens: Students may forget to use the correct ratio because they are not paying attention to the given angle.
  • How to avoid: Make sure to use the correct ratio for the given angle.
• Mistake/Trap: Squaring the wrong side in the cosine rule.
  • Why it happens: Students may square the wrong side because they are not paying attention to the given formula.
  • How to avoid: Make sure to square the correct side.
• Mistake/Trap: Using the wrong formula for the area of a triangle.
  • Why it happens: Students may use the wrong formula because they are not paying attention to the given type of triangle