Introduction to Trigonometry Exercise 8.2 — CBSE Class 10 NCERT

Understanding Trigonometric Ratios of Specific Angles

In this section of CBSE Class 10 Maths, we transition from arbitrary right triangles to specific geometric configurations. By considering standard shapes, we can calculate the precise values of trigonometric ratios for 0°, 30°, 45°, 60°, and 90°. For instance, in an isosceles right triangle with acute angles of 45°, the side ratios yield sin 45° = 1/√2 and cos 45° = 1/√2. Similarly, by bisecting an equilateral triangle, we obtain a 30°-60°-90° triangle, which directly proves why sin 30° = 1/2 and sin 60° = √3/2. Memorizing these values is critical, but understanding how they are derived geometrically provides a solid conceptual foundation that prevents silly mistakes during intense board exams. Let's learn to build the trigonometry values table logically!

Key Values of Trigonometric Ratios

sin θ values

sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1

cos θ values

cos 0° = 1, cos 30° = √3/2, cos 45° = 1/2, cos 60° = 1/2, cos 90° = 0

tan θ values

tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° is Not Defined

How to Build the Trigonometry Table Quickly in Exams

1. Write the angles and write numbers 0 to 4 — List the angles 0°, 30°, 45°, 60°, 90° in a horizontal row. Beneath them, write the integers 0, 1, 2, 3, and 4.
2. Divide by 4 and take the square root — Divide each integer by 4 (getting 0, 1/4, 2/4, 3/4, 4/4) and then calculate their square roots: √(0) = 0, √(1/4) = 1/2, √(1/2) = 1/√2, √(3/4) = √3/2, √(1) = 1. These are your sine values!
3. Reverse for Cosine — Write the sine values in reverse order to obtain the cosine values: 1, √3/2, 1/√2, 1/2, 0.
4. Divide Sine by Cosine for Tangent — Since tan θ = sin θ / cos θ, divide the sine value by the corresponding cosine value for each angle. (e.g., 0/1 = 0, and 1/0 is Not Defined).

Step-by-Step Worked Solutions for Ex 8.2

• Evaluate: sin 60° cos 30° + sin 30° cos 60°. Step 1: Substitute the values from the trigonometric table: sin 60° = √3/2, cos 30° = √3/2, sin 30° = 1/2, and cos 60° = 1/2. Step 2: Formulate the expression: (√3/2 √3/2) + (1/2 1/2). Step 3: Simplify the fractions: 3/4 + 1/4 = 4/4 = 1. Final Answer: 1.
• Evaluate: 2 tan² 45° + cos² 30° - sin² 60°. Step 1: Substitute the key values: tan 45° = 1, cos 30° = √3/2, and sin 60° = √3/2. Step 2: Plug these into the expression: 2(1)² + (√3/2)² - (√3/2)². Step 3: Simplify the terms: 2(1) + 3/4 - 3/4 = 2. Final Answer: 2.

Board Exam Tips & Pitfalls to Avoid

1. Rationalizing the Denominators: Often, your calculations will result in answers like 1/√2 or 3/√3. Always rationalize the denominator (e.g., multiply top and bottom by √3 to turn 3/√3 into √3) to secure full marks.
2. Confusing reciprocal identities: Remember that cosec 30° is the reciprocal of sin 30°, not cos 30°. Double-check your reciprocal pairs.
3. Bracket Squaring: When dealing with terms like cos² 30°, compute cos 30° first and then square the value: (√3/2)² = 3/4. Do not square the angle itself!

Practice Questions with Solutions

• Q: Evaluate: cos 45° / (sec 30° + cosec 30°) A: Step 1: Substitute the values: cos 45° = 1/√2, sec 30° = 2/√3, and cosec 30° = 2. Step 2: Express the denominator as 2/√3 + 2 = (2 + 2√3)/√3. Step 3: Solve the compound fraction: (1/√2) / ((2 + 2√3)/√3) = √3 / (√2(2 + 2√3)) = √3 / (2√2 + 2√6). Step 4: Rationalize the denominator by multiplying numerator and denominator by (2√6 - 2√2). Final answer: (3√2 - √6) / 8.
• Q: If tan(A + B) = √3 and tan(A - B) = 1/√3; 0° < A + B <= 90°; A > B, find A and B. A: Step 1: Since tan(A + B) = √3, we know that A + B = 60° (Equation 1). Step 2: Since tan(A - B) = 1/√3, we know that A - B = 30° (Equation 2). Step 3: Add Equation 1 and Equation 2: (A + B) + (A - B) = 60° + 30° => 2A = 90° => A = 45°. Step 4: Substitute A = 45° into Equation 1: 45° + B = 60° => B = 15°. Final answer: A = 45°, B = 15°.
• Q: Evaluate (5 cos² 60° + 4 sec² 30° - tan² 45°) / (sin² 30° + cos² 30°). A: Step 1: Substitute values: cos 60° = 1/2, sec 30° = 2/√3, tan 45° = 1, sin 30° = 1/2, and cos 30° = √3/2. Step 2: Note that the denominator is sin² 30° + cos² 30° = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1. Step 3: Simplify the numerator: 5(1/2)² + 4(2/√3)² - (1)² = 5(1/4) + 4(4/3) - 1 = 5/4 + 16/3 - 1. Step 4: Find the common denominator (12) for the numerator: (15 + 64 - 12)/12 = 67/12. Final answer: 67/12.
• Q: Find the value of sin 30° cos 45° + cos 30° sin 45°. A: Step 1: Identify values from the table: sin 30° = 1/2, cos 45° = 1/√2, cos 30° = √3/2, sin 45° = 1/√2. Step 2: Substitute these values into the expression: (1/2 1/√2) + (√3/2 1/`√2). Step 3: Simplify terms: 1/(2√2) + √3/(2√2) = (1 + √3)/(2√2). Step 4: Rationalize by multiplying numerator and denominator by √2: √2(1 + √3) / 4 = (√2 + √6) / 4. Final answer: (√2 + √6) / 4.

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