If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

검증된 단계별 안내

Recall the Pythagorean identity for sine and cosine: \(\sin^{2}\theta + \cos^{2}\theta = 1\).

Substitute the given value of \(\sin \theta = \frac{2\sqrt{7}}{7}\) into the identity: \(\left(\frac{2\sqrt{7}}{7}\right)^{2} + \cos^{2}\theta = 1\).

Calculate \(\sin^{2}\theta\) by squaring \(\frac{2\sqrt{7}}{7}\): \(\left(\frac{2\sqrt{7}}{7}\right)^{2} = \frac{4 \times 7}{49} = \frac{28}{49}\).

Rewrite the equation as \(\frac{28}{49} + \cos^{2}\theta = 1\) and solve for \(\cos^{2}\theta\) by subtracting \(\frac{28}{49}\) from both sides.

Since \(\theta\) is acute, take the positive square root of \(\cos^{2}\theta\) to find \(\cos \theta\).

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주요 개념

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Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental relationship allows us to find one trigonometric function if the other is known, by rearranging the equation to solve for the unknown value.

Sine Function and Its Range

The sine of an acute angle θ (0° < θ < 90°) is positive and represents the ratio of the length of the side opposite θ to the hypotenuse in a right triangle. Knowing sin θ helps determine cos θ using the Pythagorean identity.

Sign of Cosine in the First Quadrant

Since θ is acute, it lies in the first quadrant where both sine and cosine values are positive. This information is crucial when taking the square root to find cos θ, ensuring the correct positive value is chosen.