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. Identify the trigonometric function values for the quadrantal angles: \( \sin 90^\circ \) and \( \cos 180^\circ \).
2. Recall that \( \sin 90^\circ = 1 \) and \( \cos 180^\circ = -1 \).
3. Substitute these values into the expression: \(-3(\sin 90^\circ)^4 + 4(\cos 180^\circ)^3\).
4. Calculate \((\sin 90^\circ)^4 = 1^4 = 1\) and \((\cos 180^\circ)^3 = (-1)^3 = -1\).
5. Substitute these results back into the expression: \(-3(1) + 4(-1)\) and simplify.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that lie on the axes of the Cartesian coordinate system, specifically 0°, 90°, 180°, and 270°. The sine and cosine values for these angles are well-defined and can be easily memorized: sin(0°) = 0, sin(90°) = 1, cos(0°) = 1, cos(90°) = 0, cos(180°) = -1, and sin(180°) = 0. Understanding these values is crucial for evaluating trigonometric expressions involving these angles.

Trigonometric Function Values

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. For quadrantal angles, these functions yield specific values that simplify calculations. For example, sin(90°) = 1 and cos(180°) = -1. Knowing these values allows for straightforward evaluation of expressions involving trigonometric functions.

Exponentiation in Trigonometric Expressions

Exponentiation in trigonometric expressions involves raising the function values to a power. For instance, (sin 90°)⁴ means taking the sine of 90°, which is 1, and raising it to the fourth power, resulting in 1. Similarly, (cos 180°)³ involves cubing the cosine value of 180°, which is -1, yielding -1. Understanding how to manipulate these powers is essential for correctly evaluating the overall expression.