Question

Find $\mathrm{W}(\mathrm{t})$ for the given $t$. $$t=-\frac{7 \pi}{6}$$

   Find $\mathrm{W}(\mathrm{t})$ for the given $t$.
$$t=-\frac{7 \pi}{6}$$

Essentials of Precalculus

Essentials of Precalculus

Richard N. Aufmann,… 1st Edition

Chapter 4, Problem 4

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Solved by Expert Babita Kumari

07/09/2021

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We need to find the value of $W(t)$ which is equal to $(\cos(3t), \sin(3t))$. Show more…

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Find $\mathrm{W}(\mathrm{t})$ for the given $t$. $$t=-\frac{7 \pi}{6}$$

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Key Concepts

Unit Circle and Reference Angles

The unit circle is a Circle with radius one that serves as a fundamental tool in trigonometry. The position of an angle on the unit circle helps in determining the values of trigonometric functions, and the idea of a reference angle—which is the acute angle formed by the terminal side of the given angle and the horizontal axis—facilitates this calculation by allowing one to relate any angle back to an acute angle in the first quadrant. This process is essential when evaluating trigonometric expressions for angles that do not fall within the first quadrant.

Trigonometric Function Evaluation

Trigonometric function evaluation involves determining the value of functions like sine, cosine, or others at a specified angle. This typically requires understanding the definitions of these functions using the unit circle, where each angle corresponds to a particular point whose coordinates represent the values of cosine and sine. The process might include recognizing the quadrant in which the angle lies for determining the correct sign of the result.

Angle Normalization (Coterminal Angles)

Angle normalization, or finding coterminal angles, is the technique of adjusting an angle—often by adding or subtracting full rotations (2? radians in the case of radian measure)—to obtain an equivalent angle that lies within a standard interval such as [0,2?) or (-?, ?]. This is especially useful when dealing with negative angles or angles greater than 2?, as it simplifies the evaluation of trigonometric functions by relating the angle to one with a known reference value.

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