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The Pythagorean Theorem states that for a right-angled triangle 4ABC, as in figure, a2 =b2 + c2. Use this equation to prove that sin2 θ + cos2 θ = 1

Question

The Pythagorean Theorem states that for a right-angled triangle 4ABC, as in figure, a2=b2+c2 a^2 = b^2 + c^2 . Use this equation to prove that ( \sin^2 \theta + \cos^2 \theta = 1 .

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Solution

Sure, let's prove that sin²θ + cos²θ = 1 using the Pythagorean theorem.

Step 1: In a right-angled triangle, let's denote the sides as follows:

  • The side opposite to angle θ as 'b'
  • The side adjacent to angle θ as 'c'
  • The hypotenuse as 'a'

Step 2: According to the Pythagorean theorem, we have a² = b² + c².

Step 3: Now, let's express this in terms of trigonometric functions. We know that:

  • sinθ = b/a
  • cosθ = c/a

Step 4: Squaring both these equations, we get:

  • sin²θ = b²/a²
  • cos²θ = c²/a²

Step 5: Adding these two equations together, we get: sin²θ + cos²θ = b²/a² + c²/a²

Step 6: Since b²/a² + c²/a² = (b² + c²)/a², we can rewrite the equation as: sin²θ + cos²θ = (b² + c²)/a²

Step 7: But from the Pythagorean theorem, we know that (b² + c²)/a² = 1.

Therefore, we have proved that sin²θ + cos²θ = 1.

This problem has been solved

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