📚 Concepts

📚 Line of Sight

• The imaginary straight line drawn from the observer’s eye to the object being observed.

📚 Horizontal Line

• The level line from the observer’s eye that is parallel to the ground (used as a reference to measure angles).

📚 Angle of Elevation (θ)

• When an observer looks upward from the horizontal line to an object above the eye level, the angle formed is called the angle of elevation.

📚 Angle of Depression (θ)

• When an observer looks downward from the horizontal line to an object below the eye level, the angle formed is called the angle of depression.

📚 Right Triangle Formation

• In height and distance problems, the object, the observer, and the line of sight often form a right-angled triangle, allowing the use of trigonometric ratios.

📚 Trigonometric Ratios in Right Triangles

• To relate the height, distance, and angle in a right triangle, we use the basic trigonometric ratios:

  • sin θ = Opposite / Hypotenuse

  • cos θ = Adjacent / Hypotenuse

  • tan θ = Opposite / Adjacent

📚 Assumptions in Problems

• The ground is considered horizontal and level.

• The height of the observer (if given) is added to or subtracted from the object height accordingly.

• All angles are measured with respect to the horizontal.

📚 No Use of Tables/Calculators in Class 10

• CBSE restricts the use of scientific calculators or trigonometric tables students must memorize trigonometric values of standard angles like 0°, 30°, 45°, 60°, and 90°.

✍️ Formulas 

✍️ Trigonometric Ratios in a Right-Angled Triangle

• Used to relate angles with sides when solving height and distance problems:

  • 𝐬𝐢𝐧 𝜃 = 𝐎𝐩𝐩𝐨𝐬𝐢𝐭𝐞 / 𝐇𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

  • 𝐜𝐨𝐬 𝜃 = 𝐀𝐝𝐣𝐚𝐜𝐞𝐧𝐭 / 𝐇𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞

  • 𝐭𝐚𝐧 𝜃 = 𝐎𝐩𝐩𝐨𝐬𝐢𝐭𝐞 / 𝐀𝐝𝐣𝐚𝐜𝐞𝐧𝐭

✍️ Using 𝐭𝐚𝐧 𝜃 to Find Height or Distance

• 𝐇𝐞𝐢𝐠𝐡𝐭 = 𝐛 × 𝐭𝐚𝐧 𝜃 (when base or horizontal distance is known)

• 𝐁𝐚𝐬𝐞 = 𝐡 / 𝐭𝐚𝐧 𝜃 (when vertical height is known)

(Here, 𝜃 is the angle of elevation or depression)

✍️ Pythagoras Theorem (If Needed for Right Triangle)

• 𝐡² = 𝐛² + 𝐥²

(Used to calculate missing side when two sides are known)

✍️ Trigonometric Tables

• Values of standard angles (0°, 30°, 45°, 60°, 90°) are used in problems refer to the Trigonometric Table already given in the previous chapter: Introduction to Trigonometry.