Geometry Question of the Day

1. Since $\sin\theta=\frac{a}{c}$ and $\cos\theta=\frac{b}{c}$, then $\sin^2\theta+\cos^2\theta=\frac{a^2+b^2}{c^2}=1$. (correct answer)
2. Since $\sin\theta=\frac{c}{a}$ and $\cos\theta=\frac{c}{b}$, then $\sin^2\theta+\cos^2\theta=1$.
3. Since $\sin\theta=\frac{a}{b}$ and $\cos\theta=\frac{b}{c}$, then $\sin^2\theta+\cos^2\theta=1$.
4. Since $a^2+b^2=c$, dividing by $c^2$ gives $\sin^2\theta+\cos^2\theta=1$.

Explanation: This question verifies the Pythagorean identity using general triangle notation where a is opposite, b is adjacent, and c is the hypotenuse. Sine equals opposite/hypotenuse = a/c, and cosine equals adjacent/hypotenuse = b/c. Squaring these gives sin²θ = a²/c² and cos²θ = b²/c². Adding: sin²θ + cos²θ = a²/c² + b²/c² = (a² + b²)/c². By the Pythagorean Theorem, a² + b² = c², so (a² + b²)/c² = c²/c² = 1. Choice B incorrectly defines sine and cosine as c/a and c/b (reciprocals of the correct ratios). Choice C mixes up the ratios entirely. Choice D incorrectly states the Pythagorean Theorem as a² + b² = c rather than c². Careful attention to both trigonometric definitions and algebraic accuracy is essential.