Consider a triangle PQR, right angled at R, in which PQ = 29 units, QR = 21 units and ∠PQR = θ. Find the value of cos²θ − sin²θ.

Question

Accepted Answer

Let's analyze the triangle PQR, right-angled at R. Given: PQ = 29 units (hypotenuse), QR = 21 units (one leg), and ∠PQR = θ. First, let's find PR (the other leg) using the Pythagoras theorem:

PQ² = PR² + QR²
29² = PR² + 21²
841 = PR² + 441
PR² = 841 - 441 = 400
PR = √400 = 20 units

Now, at angle θ = ∠PQR, QR is the side adjacent to θ, and PR is the side opposite to θ (since R is the right angle, and PQ is the hypotenuse).

So,
cosθ = adjacent/hypotenuse = QR/PQ = 21/29
sinθ = opposite/hypotenuse = PR/PQ = 20/29

We are asked to find cos²θ − sin²θ:
cos²θ − sin²θ = (cosθ + sinθ)(cosθ − sinθ)
But let's compute directly:
cos²θ − sin²θ = (21/29)² − (20/29)² = (441/841) − (400/841) = (441 − 400)/841 = 41/841

Wait, this matches option 4, but let's double-check the assignment of sides:
At ∠PQR = θ, QR is adjacent, PR is opposite, PQ is hypotenuse. So the calculation is correct.

But the calculation above gives 41/841, which is option 4. However, let's check the question's options and the calculation again.

cos²θ − sin²θ = (21/29)² − (20/29)² = (441 − 400)/841 = 41/841

So the correct answer is 41/841 (option 4). However, the answer key in the options marks 20/841 as correct. Let's check if there is a misinterpretation.

Let's check if the triangle is drawn differently. If QR = 21, PQ = 29, and right angle at R, then PR = 20.
At angle θ = ∠PQR, adjacent = QR = 21, opposite = PR = 20, hypotenuse = PQ = 29.

So,
cosθ = 21/29
sinθ = 20/29
cos²θ − sin²θ = (21/29)² − (20/29)² = (441 − 400)/841 = 41/841

Thus, the correct answer is 41/841.

Final answer: Option 4.

Answer

Answer

Answer