Introduction to Trigonometry — Class 10

Trigonometric ratios • Standard identities • Heights & distances • Practice
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2. Basic Trigonometric Ratios (Definitions)

For an acute angle θ in a right triangle:

sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent

Only three independent ratios; others are derived (next section).

3. Reciprocal Ratios

cosec θ = 1/sin θ = hypotenuse/opposite
sec θ = 1/cos θ = hypotenuse/adjacent
cot θ = 1/tan θ = adjacent/opposite

Memorise reciprocal pairs: (sin, cosec), (cos, sec), (tan, cot).

4. Standard Identities

Important identities to remember:

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Use these to convert between ratios and simplify expressions.

5. Values at Standard Angles

Common angle values (θ in degrees): 0°, 30°, 45°, 60°, 90°. Remember symmetry and signs over quadrants; for class 10 we focus on acute angles.

θsin θcos θtan θ
0°010
30°1/2√3/21/√3
45°√2/2√2/21
60°√3/21/2√3
90°10—

6. Heights and Distances (Applications)

These problems use right-triangle trigonometry to find heights or distances where direct measurement is hard. Steps:

  1. Draw a clear diagram and label known distances and angles.
  2. Identify the right triangle(s) and the angle θ used.
  3. Choose appropriate ratio (sin/cos/tan) and set up equation.
  4. Solve for required height/distance and check units.

Example. From point P on ground, angle of elevation to top of tower is 30°. If distance from P to foot of tower is 50 m, height h = 50 × tan 30° = 50 × (1/√3) ≈ 28.87 m.

7. Solved Examples

Example 1. In right triangle, if sin θ = 3/5, find cos θ and tan θ.

Use sin²θ + cos²θ = 1 ⇒ cos θ = √(1 − 9/25) = √(16/25) = 4/5. tan θ = sin/cos = (3/5)/(4/5) = 3/4.

Example 2. A ladder leans against a wall making 60° with ground and reaches 10 m up the wall. Find length of ladder.

cos 60° = adjacent/hypotenuse = 10 / ladder ⇒ 1/2 = 10 / L ⇒ L = 20 m.

Example 3 (Heights & Distances). From two points A and B, 30 m apart on level ground, angles of elevation to top of building are 30° and 45° respectively. Find height of building.

Let foot be C, distances from C to A and B be x and x+30. Use tan relations: h/x = tan 30°, h/(x+30)=tan45°=1. Solve: h = x/√3 and h = x+30 ⇒ x/√3 = x+30 ⇒ solve for x then h.

8. Practice Questions

  1. If cos θ = 5/13, find sin θ and tan θ (θ acute).
  2. From a point 40 m away from a tower, angle of elevation to top is 37°. Find tower height (use tan 37° ≈ 0.75).
  3. A flagpole at top of a building subtends an angle of elevation 30° at a point 50 m away from the base. Find height of flagpole if building height is 20 m.
  4. Show that tan θ = sin θ / cos θ using definitions.

9. Exam Tips & Quick Memory Aids

  • Remember SOH-CAH-TOA (Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj) for quick recall.
  • Sketch diagrams carefully — mistakes in labelling cause most errors.
  • Use identities to find missing ratios; check signs for angles beyond class 10 scope.
  • For heights & distances, ensure angles of elevation/depression are treated consistently (elevation from ground up; depression from observer down).