Introduction to Trigonometry – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

Trigonometric ratios relate the sides of a right triangle to its acute angles. Standard angle values and three identities solve most questions.

Ratios (for angle A in a right triangle)

• sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent.
• cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A.
• tan A = sin A / cos A.

Standard values

• sin 0°, 30°, 45°, 60°, 90° = 0, 1/2, 1/√2, √3/2, 1.
• cos is the same list in reverse order.
• tan 0°, 30°, 45°, 60° = 0, 1/√3, 1, √3; tan 90° is not defined.

Identities

• sin²A + cos²A = 1.
• 1 + tan²A = sec²A (A ≠ 90°).
• 1 + cot²A = cosec²A (A ≠ 0°).

Key terms

Hypotenuse

the side opposite the right angle (longest side)

Trigonometric identity

an equation true for every angle where both sides are defined

Complementary angles

two angles whose sum is 90°, e.g. sin(90° − A) = cos A

Formulas

• sin²A + cos²A = 1
• sec²A − tan²A = 1
• cosec²A − cot²A = 1

Important questions with answers

Quick revision

• sin = P/H, cos = B/H, tan = P/B.
• sin 30° = cos 60° = 1/2, tan 45° = 1.
• Three identities: sin² + cos² = 1, sec² − tan² = 1, cosec² − cot² = 1.

Exam tips

• Draw the triangle and use Pythagoras to find the third side when one ratio is given.
• Learn the table with the √n/2 trick: sin 0°…90° = √0/2, √1/2, √2/2, √3/2, √4/2.

Practice MCQs

1. sin 30° equals

   1. 1
   2. 1/2
   3. √3/2
   4. 1/√2
   Show answer

   Answer: (b) 1/2. Standard value.

2. tan 45° equals

   1. 0
   2. 1
   3. √3
   4. not defined
   Show answer

   Answer: (b) 1. Opposite = adjacent at 45°.

3. sin²θ + cos²θ =

   1. 0
   2. tan²θ
   3. 2
   4. 1
   Show answer

   Answer: (d) 1. Basic identity.

4. If tan A = 3/4, then sin A =

   1. 3/4
   2. 4/5
   3. 3/5
   4. 5/3
   Show answer

   Answer: (c) 3/5. Hypotenuse = 5, so sin A = 3/5.

5. The value of 2 tan 30° / (1 + tan²30°) is

   1. tan 60°
   2. cos 60°
   3. sin 60°
   4. sin 30°
   Show answer

   Answer: (c) sin 60°. 2(1/√3)/(4/3) = √3/2 = sin 60°.

6. (1 + tan²A) cos²A =

   1. sin²A
   2. 0
   3. 1
   4. sec A
   Show answer

   Answer: (c) 1. sec²A × cos²A = 1.

7. If sec θ + tan θ = p, then sec θ − tan θ =

   1. −p
   2. p
   3. 1/p
   4. p²
   Show answer

   Answer: (c) 1/p. (sec θ + tan θ)(sec θ − tan θ) = sec²θ − tan²θ = 1.

8. If sin(A − B) = 1/2 and cos(A + B) = 1/2 (0° < A + B ≤ 90°), then A =

   1. 45°
   2. 30°
   3. 60°
   4. 15°
   Show answer

   Answer: (a) 45°. A − B = 30°, A + B = 60° ⇒ A = 45°.