Introduction to Trigonometry — Class 10 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Introduction to Trigonometry" — 9 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Introduction to Trigonometry" — 9 important questions with detailed answers for CB…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

Define the six trigonometric ratios for a right triangle. State the trigonome…
In a right triangle ABC with right angle at B, AB = 5 cm and AC = 13 cm. Find…
Prove the identity: sin²θ + cos²θ = 1
Simplify: (sin 30° + cos 60°)/(tan 45°) + sec 60° - cosec 30°
If sin θ = 3/5 and θ is acute, find cos θ and tan θ.
If tan θ = 5/12 and θ is in the first quadrant, find sin θ and cos θ.
+ 3 more questions in the full chapter

Solutions Summary:

Question	Status
Define the six trigonometric ratios for a right triangle.…	✓ Solved
In a right triangle ABC with right angle at B, AB = 5 cm …	✓ Solved
Prove the identity: sin²θ + cos²θ = 1	✓ Solved
Simplify: (sin 30° + cos 60°)/(tan 45°) + sec 60° - cosec…	✓ Solved
If sin θ = 3/5 and θ is acute, find cos θ and tan θ.	✓ Solved
If tan θ = 5/12 and θ is in the first quadrant, find sin …	✓ Solved

Showing 6 of 9 questions

Q1: Define the six trigonometric ratios for a right triangle. State the trigonometric values for the standard angles 0°, 30°, 45°, 60°, and 90°.

Trigonometric Ratios (for a right triangle with reference angle θ): Let a right triangle have: - Opposite side (opposite to angle θ) = opposite - Adjacent side (adjacent to angle θ) = adjacent - Hypotenuse = hypotenuse The six ratios are: 1. sin θ = opposite / hypotenuse 2. cos θ = adjacent / hypotenuse 3. tan θ = opposite / adjacent 4. cot θ = adjacent / opposite = 1/tan θ 5. sec θ = hypotenuse / adjacent = 1/cos θ 6. cosec θ = hypotenuse / opposite = 1/sin θ MNEMONIC: SOH-CAH-TOA Sin=Opposi...

Q2: In a right triangle ABC with right angle at B, AB = 5 cm and AC = 13 cm. Find all six trigonometric ratios of angle A.

Given: Right triangle ABC with ∠B = 90° AB = 5 cm (side adjacent to angle A) AC = 13 cm (hypotenuse) Find: All six trigonometric ratios of angle A Step 1: Find BC (the side opposite to angle A) using Pythagorean theorem. AC² = AB² + BC² 13² = 5² + BC² 169 = 25 + BC² BC² = 144 BC = 12 cm Step 2: Identify the sides relative to angle A. Opposite side (opposite to ∠A) = BC = 12 cm Adjacent side (adjacent to ∠A) = AB = 5 cm Hypotenuse = AC = 13 cm Step 3: Calculate sin A. sin A = opposite / hypot...

Q3: Prove the identity: sin²θ + cos²θ = 1

Prove: sin²θ + cos²θ = 1 Proof: Consider a right triangle with: - Angle θ at one vertex - Opposite side = a - Adjacent side = b - Hypotenuse = c Step 1: Write the definitions of sin θ and cos θ. sin θ = a/c cos θ = b/c Step 2: Square both ratios. sin²θ = a²/c² cos²θ = b²/c² Step 3: Add the squared ratios. sin²θ + cos²θ = a²/c² + b²/c² sin²θ + cos²θ = (a² + b²)/c² Step 4: Apply the Pythagorean theorem. In a right triangle: a² + b² = c² Substitute: sin²θ + cos²θ = c²/c² sin²θ + cos²θ = 1 He...

Q4: Simplify: (sin 30° + cos 60°)/(tan 45°) + sec 60° - cosec 30°

Given: Simplify (sin 30° + cos 60°)/(tan 45°) + sec 60° - cosec 30° Step 1: Recall the values of trigonometric functions for standard angles. sin 30° = 1/2 cos 60° = 1/2 tan 45° = 1 sec 60° = 2 cosec 30° = 2 Step 2: Substitute these values. = (1/2 + 1/2)/(1) + 2 - 2 Step 3: Simplify the first term. = (1)/(1) + 2 - 2 = 1 + 2 - 2 = 1 Answer: The simplified expression equals 1

Q5: If sin θ = 3/5 and θ is acute, find cos θ and tan θ.

Given: sin θ = 3/5, θ is acute (0° < θ < 90°) Find: cos θ and tan θ Step 1: Use the Pythagorean identity sin²θ + cos²θ = 1. sin²θ + cos²θ = 1 (3/5)² + cos²θ = 1 9/25 + cos²θ = 1 cos²θ = 1 - 9/25 cos²θ = 25/25 - 9/25 cos²θ = 16/25 Step 2: Take the square root. cos θ = ±4/5 Since θ is acute, cos θ > 0. cos θ = 4/5 Step 3: Find tan θ. tan θ = sin θ / cos θ tan θ = (3/5) / (4/5) tan θ = (3/5) × (5/4) tan θ = 3/4 Answer: cos θ = 4/5 = 0.8 tan θ = 3/4 = 0.75 Verification: sin²θ + cos²θ...

Q6: If tan θ = 5/12 and θ is in the first quadrant, find sin θ and cos θ.

Given: tan θ = 5/12, θ is in the first quadrant Find: sin θ and cos θ Step 1: Understand what tan θ = 5/12 means. tan θ = opposite / adjacent = 5/12 This means in a right triangle: Opposite side = 5 Adjacent side = 12 Step 2: Find the hypotenuse using Pythagorean theorem. hypotenuse² = opposite² + adjacent² hypotenuse² = 5² + 12² hypotenuse² = 25 + 144 hypotenuse² = 169 hypotenuse = 13 Step 3: Find sin θ. sin θ = opposite / hypotenuse = 5/13 Step 4: Find cos θ. cos θ = adjacent / hypotenus...

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