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Difficulty: MediumBasic Trigonometric Ratios, Special Angles, and Identities

If θ\theta is an acute angle such that sin⁡θ=513\sin \theta = \frac{5}{13}, what is the exact numerical value of 169(sin⁡2θ−cos⁡2θ)169(\sin^2 \theta - \cos^2 \theta)?

Answer: -119

Answer

The exact numerical value of the expression is −119-119.
Using the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, we find cos⁡2θ=1−25169=144169\cos^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}. Then sin⁡2θ−cos⁡2θ=25−144169=−119169\sin^2 \theta - \cos^2 \theta = \frac{25 - 144}{169} = -\frac{119}{169}. Multiplying by 169169 gives −119-119.

Step-by-Step Solution

1
Calculate cos⁡2θ\cos^2 \theta using the Pythagorean trigonometric identity.
cos⁡2θ=144169\cos^2 \theta = \frac{144}{169}
Since sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, subtracting sin⁡2θ=25169\sin^2 \theta = \frac{25}{169} from 11 yields 144169\frac{144}{169}.
2
Compute the difference sin⁡2θ−cos⁡2θ\sin^2 \theta - \cos^2 \theta.
sin⁡2θ−cos⁡2θ=−119169\sin^2 \theta - \cos^2 \theta = -\frac{119}{169}
Subtracting 144169\frac{144}{169} from 25169\frac{25}{169} gives −119169-\frac{119}{169}.
3
Scale the difference by 169169.
169×(−119169)=−119169 \times \left(-\frac{119}{169}\right) = -119
Multiplying −119169-\frac{119}{169} by 169169 cancels the denominator, leaving −119-119.

Key Concept

Pythagorean Trigonometric Identity
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Topics

Basic Trigonometric Ratios, Special Angles, and Identities

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