Introduction to Trigonometry Class 10 Maths Worksheet with Answers Chapter 8

Introduction to Trigonometry is yet another maths chapter in Class 10 mathematics that is going to be with you for long. This curated questions in this worksheet is selected so that you could be able to spot the correct ratio and solve confidently without confusion. It is split into four levels: Basic (ratio setup + definitions), Standard (direct evaluations and routine sums), Advance (identity-based simplification), and HOTS (reasoning and tricky ones).

Topics this chapter covers

• Right triangle ratios
• Trig ratios definitions
• Standard angle values
• Identities and simplification
• Heights and distances

Class 10 Maths Worksheet – Chapter 8: Introduction to Trigonometry

Basic

1. Write the meaning of trigonometry (in 1 line).
2. In a right ΔABC, ∠B = 90° and ∠A is acute.
   Fill the blanks:
   • Opposite to ∠A = ______
   • Adjacent to ∠A = ______
   • Hypotenuse = ______
3. If in a right triangle (for an acute angle θ), sin θ = 3/5, find cos θ.
4. If tan θ = 4/3 (θ acute), find sin θ and cos θ.
5. True/False (write 1 line reason):
   • sin θ is always greater than 1.
   • cos 0° = 1.
   • tan 90° is defined.
6. Find the value:
   (1 + sin θ)(1 − sin θ) in simplest form.
7. If sec θ = 5/4 (θ acute), find tan θ.

Standard

1. Using standard values, find:
   sin 30° + cos 60° + tan 45°
2. Evaluate:
   sec²30° − tan²30°
3. If cot θ = 12/5 (θ acute), find all other trigonometric ratios of θ.
4. A right ΔPQR is right-angled at Q. If PQ = 6 and PR = 10, find:
   sin R, cos R, and tan R.
5. If sin θ = 1/2 and θ is acute, find θ (in degrees).
6. Verify (by simplifying LHS):
   (1 + tan²θ) / sec²θ = 1
7. Show that:
   sin²θ + cos²θ = 1
   (Write 3–4 clear steps.)

Advance

1. If tan θ = 1/√3 and θ is acute, find:
   sin θ and cos θ.
2. Prove:
   (cosec θ − cot θ)(cosec θ + cot θ) = 1
3. If sin θ = 3/5 (θ acute), find:
   sec θ and cosec θ.
4. Simplify:
   (1 − cos²θ) / sin²θ
5. Solve for acute θ:
   sec θ = cosec θ
6. In a right triangle, one acute angle is 60°. If hypotenuse = 14 cm, find the other two sides.
   (Answer in surd form.)
7. Prove:
   (1 + sin θ) / cos θ = sec θ + tan θ

HOTS

1. Without using a calculator, arrange in increasing order:
   sin 30°, sin 45°, sin 60°
   (Write a 2-line reason.)
2. A student says: “Because sin θ increases as θ increases, tan θ must also always increase in the same way.”
   Is the conclusion correct for 0° to 90°? Write a clear 3–4 line explanation.
3. If sin θ = cos θ and θ is acute, find θ and justify in 2 lines.
4. Choose the correct statement and justify:
   • A) sec²θ − tan²θ = 1 for all θ including 90°
   • B) sec²θ − tan²θ = 1 for 0° ≤ θ < 90°
5. If tan θ = 2 (θ acute), find the exact value of:
   (1 − tan²θ) / (1 + tan²θ)
6. A right triangle has sin θ = 0.6 (θ acute). Another triangle has sin θ = 3/5.
   Are the θ values necessarily equal? Answer Yes/No with reason.
7. Explain in 2–3 lines:
   Why are cosec 0° and sec 90° not defined?

Answer Key

Basic – Answers

1. Ans: Trigonometry studies relationships between sides and angles of triangles.
   Hint: Mainly for right triangles in this chapter.
2. Ans: Opposite = BC, Adjacent = AB, Hypotenuse = AC.
   Hint: Hypotenuse is opposite the 90° angle.
3. Ans: cos θ = 4/5.
   Hint: Use cos²θ = 1 − sin²θ.
4. Ans: sin θ = 4/5, cos θ = 3/5.
   Hint: Take opposite:adjacent = 4:3, then hypotenuse = 5 (3–4–5).
5. Ans:
   • False (sin θ ≤ 1)
   • True
   • False (tan 90° not defined)

   Hint: Division by 0 makes a ratio undefined.

6. Ans: cos²θ.
   Hint: (1 + a)(1 − a) = 1 − a².
7. Ans: tan θ = 3/4.
   Hint: sec²θ = 1 + tan²θ ⇒ tan²θ = (5/4)² − 1 = 9/16.

Standard – Answers

1. Ans: sin 30° + cos 60° + tan 45° = 1/2 + 1/2 + 1 = 2.
   Hint: Use standard angle values.
2. Ans: sec²30° − tan²30° = 1.
   Hint: Identity: sec²θ − tan²θ = 1.
3. Ans:
   • cot θ = 12/5 ⇒ tan θ = 5/12
   • Take opposite = 5, adjacent = 12 ⇒ hypotenuse = 13
   • sin θ = 5/13, cos θ = 12/13
   • cosec θ = 13/5, sec θ = 13/12

   Hint: Use a right-triangle ratio model.

4. Ans: QR = √(10² − 6²) = √(100 − 36) = 8.
   sin R = PQ/PR = 6/10 = 3/5, cos R = QR/PR = 8/10 = 4/5, tan R = PQ/QR = 6/8 = 3/4.
   Hint: First find the missing side using Pythagoras.
5. Ans: θ = 30°.
   Hint: sin 30° = 1/2 (acute angle).
6. Ans: 1
   Hint: sec²θ = 1 + tan²θ ⇒ (1 + tan²θ)/sec²θ = 1.
7. Ans: sin²θ + cos²θ = 1.
   Hint: Divide Pythagoras relation by hypotenuse².

Advance – Answers

1. Ans: θ = 30° ⇒ sin θ = 1/2, cos θ = √3/2.
   Hint: tan 30° = 1/√3.
2. Ans: (cosec²θ − cot²θ) = 1.
   Hint: Use (a − b)(a + b) = a² − b² and identity cosec²θ = 1 + cot²θ.
3. Ans: sec θ = 5/4 and cosec θ = 5/3.
   Hint: If sin θ = 3/5 then cos θ = 4/5.
4. Ans: 1
   Hint: 1 − cos²θ = sin²θ.
5. Ans: θ = 45°.
   Hint: sec θ = cosec θ ⇒ cos θ = sin θ ⇒ θ = 45° (acute).
6. Ans: Adjacent = 7, Opposite = 7√3.
   Hint: sin 60° = (opposite)/(hypotenuse) = √3/2.
7. Ans: True.
   Hint: RHS = 1/cosθ + sinθ/cosθ = (1 + sinθ)/cosθ.

HOTS – Answers

1. Ans: sin 30° < sin 45° < sin 60°.
   Hint: Values are 1/2, 1/√2, √3/2.
2. Ans: The conclusion is broadly correct on 0° to 90°, but the reason must be careful.
   Hint: tan θ = sin θ / cos θ; sin increases and cos decreases, so tan increases faster.
3. Ans: θ = 45°.
   Hint: In acute angles, sin θ = cos θ only at 45°.
4. Ans: B is correct.
   Hint: At 90°, tan θ and sec θ are not defined.
5. Ans: (1 − 4)/(1 + 4) = −3/5.
   Hint: Substitute tan²θ = 4.
6. Ans: Yes.
   Hint: 0.6 = 3/5, so both represent the same value of sin θ (acute).
7. Ans: cosec 0° = 1/sin 0° and sec 90° = 1/cos 90°.
   Since sin 0° = 0 and cos 90° = 0, division by 0 occurs.
   Hint: Undefined means “cannot divide by zero”.

Worksheet for Other chapters

• Real Numbers Class 10 Maths Worksheet
• Polynomials Class 10 Maths Worksheet
• Pair of Linear Equations in Two Variables Class 10 Maths Worksheet
• Quadratic Equations Class 10 Maths Worksheet
• Arithmetic Progressions Class 10 Maths Worksheet
• Some Applications of Trigonometry Class 10 Maths Worksheet
• Areas Related to Circles Class 10 Maths Worksheet
• Probability Class 10 Maths Worksheet