Problem 14 - Algebra and Trigonometry

Question

evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. $$ \cos \frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

\(\cos(\frac{3 \pi }{2}) = 0\).

Step 1: Identify quadrantal angle

The given value is \(\frac{3 \pi}{2}\). This is a quadrantal angle, so the cosine function will simplify to either 0, 1, -1, or be undefined.

Step 2: Evaluate Cosine at given value

To evaluate the cosine at \(\frac{3 \pi }{2}\), recall that cosine corresponds to the x-coordinate in the unit circle. At \(\frac{3 \pi }{2}\) (or 270 degrees), the position on the unit circle is in the negative y-direction, where the x-coordinate is 0.

Step 3: State the result

Therefore, \(\cos(\frac{3 \pi }{2}) = 0\).

Key Concepts

Quadrantal Angles

Quadrantal angles are special angles used frequently when dealing with trigonometric functions. They lie at the intersection of the axes on the coordinate plane. These angles are multiples of \(\pi/2\) radians or 90 degrees. When you convert these to degrees, they are 0, 90, 180, or 270 degrees. These angles are significant because evaluating trigonometric functions at these points often results in values of -1, 0, or 1, making calculations more straightforward. Consider the quadrantal angles along with the unit circle. They help in determining precise trigonometric values and remove complexities that come with angles that are not quadrantal. It’s always either on the x-axis or y-axis, simplifying most trigonometric evaluations.

Unit Circle

The unit circle is a crucial concept in trigonometry. It's a circle with a radius of 1, centered at the origin of the coordinate plane. Each point on the unit circle corresponds to an angle that has a specific sine and cosine value. This is why the unit circle is a valuable tool for understanding trigonometric functions.

• The x-coordinate of any point on the unit circle reflects the cosine of the angle.
• The y-coordinate represents the sine of the angle.
• The entire unit circle is divided into four quadrants, each equivalent to a range of angles from 0 to \(2\pi\).

When dealing with cosine at specific angles, such as \(\frac{3\pi}{2}\), imagine a line going from the center through the point representing the angle. This point will directly give you the cosine value based on its x-coordinate.

Cosine Function

The cosine function is one of the primary trigonometric functions. It relates an angle of a right triangle to the ratio of the adjacent side length to its hypotenuse.For angles on the unit circle, the cosine value is equivalent to the x-coordinate of the point where the terminal side of the angle intersects the circle. Let's dissect some key aspects:

• Cosine values range between -1 and 1.
• At quadrantal angles, these values tend to be -1, 0, or 1, making calculations easier.
• The cosine of \(0\) degrees or \(0\) radians is \(1\), while the cosine of \(180\) degrees or \(\pi\) radians is \(-1\).
• Notably, at \(\frac{3\pi}{2}\) (or 270 degrees), the x-coordinate is 0. Hence, \(\cos \frac{3\pi}{2} = 0\).

Understanding cosine in the context of the unit circle aids in quickly resolving trigonometric functions, especially at quadrantal angles.