Consider △ACB, right-angled at C, in which AB=29 units, BC=21 units and ∠ABC=θ. Determine the values of (i) cos2θ+sin2θ, (ii) cos2θ−sin2θ.
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Step-by-Step Solution
Step 1: Identify given values and the unknown side
We are given a right-angled triangle △ACB, where the right angle is at C. We know the lengths of the hypotenuse AB=29 units and the side BC=21 units. We need to find the length of the side AC to calculate the trigonometric ratios. We can use the Pythagorean theorem for this.
Step 2: Calculate the length of side AC
Substitute the given values into the Pythagorean theorem to calculate the length of side AC. We have AB=29 and BC=21. Therefore, AC=292−212=841−441=400=20 units. This value will be used to determine the sine and cosine of θ.
Step 3: Determine sinθ and cosθ
For angle θ (which is ∠ABC), the side opposite is AC, and the side adjacent is BC. The hypotenuse is AB. Using the definitions of sine and cosine for a right-angled triangle, we find sinθ=ABAC=2920 and cosθ=ABBC=2921. These values are crucial for solving both parts of the problem.
Step 4: Calculate cos2θ+sin2θ
Substitute the calculated values of sinθ and cosθ into the expression cos2θ+sin2θ. This fundamental trigonometric identity states that for any angle θ, cos2θ+sin2θ=1. Our calculation confirms this identity for the given triangle.
Step 5: Calculate cos2θ−sin2θ
Substitute the calculated values of sinθ and cosθ into the expression cos2θ−sin2θ. Perform the squaring and subtraction operations to find the final numerical value. This gives us 84141.