Use trigonometric function values of quadrantal angles to evaluate each expression. (sin 180°)² + (cos 180°)²

검증된 단계별 안내

Recall the Pythagorean identity in trigonometry: \(\sin^2 \theta + \cos^2 \theta = 1\) for any angle \(\theta\).

Identify the angle given in the problem, which is \(180^\circ\), a quadrantal angle lying on the negative x-axis.

Find the exact values of \(\sin 180^\circ\) and \(\cos 180^\circ\) using the unit circle: \(\sin 180^\circ = 0\) and \(\cos 180^\circ = -1\).

Substitute these values into the expression: \((\sin 180^\circ)^2 + (\cos 180^\circ)^2 = (0)^2 + (-1)^2\).

Simplify the expression by squaring each term and adding them together to verify the identity.

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Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis in the coordinate plane, typically 0°, 90°, 180°, 270°, and 360°. Their sine and cosine values are either 0, 1, or -1, which simplifies calculations and helps evaluate trigonometric expressions easily.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, (sin θ)² + (cos θ)² = 1. This fundamental relationship holds true for all angles, including quadrantal angles, and is essential for verifying or simplifying trigonometric expressions.

Evaluating Trigonometric Functions at Specific Angles

Evaluating sine and cosine at specific angles involves substituting the angle into the function and using known values or the unit circle. For example, sin 180° = 0 and cos 180° = -1, which can be directly applied to simplify expressions.