Use trigonometric function values of quadrantal angles to evaluate each expression. tan 360° + 4 sin 180° + 5(cos 180°)²

Verified step by step guidance

1. Recall the values of the trigonometric functions at quadrantal angles: \( \tan 360^\circ = 0 \), \( \sin 180^\circ = 0 \), and \( \cos 180^\circ = -1 \).
2. Substitute these values into the expression: \( \tan 360^\circ + 4 \sin 180^\circ + 5 (\cos 180^\circ)^2 = 0 + 4 \times 0 + 5 \times (-1)^2 \).
3. Simplify the powers and multiplication: \( 5 \times (-1)^2 = 5 \times 1 = 5 \).
4. Add all the terms together: \( 0 + 0 + 5 \).
5. The final step is to sum these values to get the result of the expression.

Key Concepts

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 trigonometric function values are special and often simple, such as sine or cosine being 0, ±1. Understanding these values helps evaluate expressions involving these angles directly.

Trigonometric Function Values at Specific Angles

Knowing the exact sine, cosine, and tangent values at key angles like 180° and 360° is essential. For example, sin 180° = 0, cos 180° = -1, and tan 360° = 0. These values allow substitution into expressions to simplify and evaluate them accurately.

Evaluating Expressions with Powers and Sums of Trigonometric Functions

When expressions involve powers, such as (cos 180°)², and sums of multiple trigonometric terms, it is important to apply the correct order of operations and substitute values carefully. Squaring a trigonometric value affects the sign and magnitude, influencing the final result.