Use trigonometric function values of quadrantal angles to evaluate each expression. ―3(sin 90°)⁴ + 4(cos 180°)³

Verified step by step guidance

1. Recall the values of sine and cosine at quadrantal angles: \( \sin 90^\circ = 1 \) and \( \cos 180^\circ = -1 \).
2. Evaluate \( (\sin 90^\circ)^4 \) by raising \( \sin 90^\circ = 1 \) to the 4th power: \( 1^4 \).
3. Evaluate \( (\cos 180^\circ)^3 \) by raising \( \cos 180^\circ = -1 \) to the 3rd power: \( (-1)^3 \).
4. Substitute these values back into the expression: \( -3 \times (1^4) + 4 \times (-1)^3 \).
5. Simplify the expression step-by-step by performing the multiplications and additions to find the final value.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles located on the x- or y-axis in the coordinate plane, specifically 0°, 90°, 180°, 270°, and 360°. Their sine and cosine values are always 0, ±1, which simplifies trigonometric calculations significantly.

Trigonometric Function Values at Quadrantal Angles

The sine and cosine values at quadrantal angles are fixed: sin 90° = 1, cos 180° = -1, etc. Knowing these exact values allows direct substitution into expressions without approximation, making evaluation straightforward.

Exponentiation of Trigonometric Values

Raising sine or cosine values to powers involves multiplying the value by itself repeatedly. For example, (sin 90°)⁴ means (1)⁴ = 1. Understanding how powers affect ±1 and 0 is crucial for correctly simplifying expressions.