Question 21 of 50beginner🔧 ApplyShort Answer2 marks

Prove that: 2sin² 3π/4 + 2 cos² π/4 + 2 sec² π/3 = 10

Correct Answer

L.H.S. = 2sin²(3π/4) + 2cos²(π/4) + 2sec²(π/3)

First, we find sin(3π/4). Since 3π/4 = π - π/4, we have sin(3π/4) = sin(π - π/4) = sin(π/4) = 1/2\sqrt{2}.

From the context, we know: cos(π/4) = 1/2\sqrt{2} and sec(π/3) = 2.

Now substituting these values:

  • 2sin²(3π/4) = 2 × (1/2\sqrt{2})² = 2×12 \times 1/2 = 1
  • 2cos²(π/4) = 2 × (1/2\sqrt{2})² = 2×12 \times 1/2 = 1
  • 2sec²(π/3) = 2 × (2)² = 2×42 \times 4 = 8

Therefore, L.H.S. = 1 + 1 + 8 = 10 = R.H.S.

Hence proved.

Exercise: EXERCISE 3.3 | Q: 4 | (Page 25)
For More Understanding

Explanation

This question from Exercise 3.3 tests students on evaluating trigonometric functions at specific angles. The key steps involve: (1) recognizing that sin(3π/4) = sin(π - π/4) = sin(π/4) using the identity shown in Example 10, (2) using standard values cos(π/4) = 1/2\sqrt{2} from Example 11, and (3) using sec(π/3) = 2 from Example 10. The question is worth 2 marks as it requires multiple substitutions and calculations.

Solution Steps

  1. Step 1: Express sin(3π/4) = sin(π - π/4) = sin(π/4) = 1/2\sqrt{2}

  2. Step 2: Use standard values: cos(π/4) = 1/2\sqrt{2} and sec(π/3) = 2

  3. Step 3: Calculate 2sin²(3π/4) = 2 × (1/2\sqrt{2})² = 1

  4. Step 4: Calculate 2cos²(π/4) = 2 × (1/2\sqrt{2})² = 1

  5. Step 5: Calculate 2sec²(π/3) = 2×42 \times 4 = 8

  6. Step 6: Add all terms: 1 + 1 + 8 = 10 = R.H.S.

PreviousNext