Problem 26 - Precalculus : Building Concepts and Connections

Question

Find the exact values of all the trigonometric functions for the giocn calues of \(t .\) If a certain value is undefined, state sa Do not use a calculator. $$t=-\frac{3 \pi}{2}$$

Step-by-Step Solution

Answer

The exact values for the trigonometric functions are: \(sin(t) = -1\), \(cos(t) = 0\), \(tan(t)\) is undefined, \(csc(t) = -1\), \(sec(t)\) is undefined, \(cot(t) = 0\).

Step 1: Evaluate sin(t)

We start with sine, for \(t=-\frac{3\pi}{2}\), we know \(sin(t)=-1\). As, at this point on the unit circle the y-coordinate is -1.

Step 2: Evaluate cos(t)

Next, we determine cosine, for \(t=-\frac{3\pi}{2}\), \(cos(t)=0\). As, at this point on the unit circle, the x-coordinate is 0.

Step 3: Evaluate tan(t)

Now, determine tangent, \(tan(t) = \frac{sin(t)}{cos(t)} = \frac{-1}{0}\). However, division by zero is undefined, so \(tan(t)\) is undefined.

Step 4: Evaluate csc(t)

We determine cosecant, \(csc(t) = \frac{1}{sin(t)} = \frac{1}{-1} = -1\).

Step 5: Evaluate sec(t)

We determine secant, \(sec(t) = \frac{1}{cos(t)} = \frac{1}{0}\). Division by zero is undefined, so \(sec(t)\) is undefined.

Step 6: Evaluate cot(t)

We determine cotangent, \(cot(t) = \frac{cos(t)}{sin(t)} = \frac{0}{-1} = 0\).

Key Concepts

Unit Circle and Trigonometry

When we talk about the unit circle in trigonometry, we're referring to a circle with a radius of one unit centered at the origin (0,0) of a coordinate system. Understanding the unit circle is critical as it provides a graphical representation of the sine and cosine functions. In fact, for any angle \( t \), the coordinates \( (x,y) \) of the point where the terminal side of the angle intersects the unit circle are equivalent to \( (cos(t), sin(t)) \).

For example, with an angle \( t = -\frac{3 \pi}{2} \), the terminal side lies along the negative y-axis. This explains why in the first step of our solution, \( sin(t) = -1 \), because the y-coordinate of the intersection point is -1. Similarly, the x-coordinate at that point is 0, which is why \( cos(t) = 0 \) in the second step of our solution.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that measure the y and x coordinates on the unit circle, respectively. For any angle \( t \), \( sin(t) \) represents the vertical distance from the origin to the terminal side of the angle, while \( cos(t) \) represents the horizontal distance.

The functions have a range of -1 to 1, as they correspond to the coordinates of a point on the unit circle. This is why when we evaluated \( sin(t) \) at \( t = -\frac{3 \pi}{2} \), the value is -1, falling within this range. The fact that \( cos(t) = 0 \) reinforces that at this angle, the point lies directly on the y-axis, having no horizontal distance from the origin.

Tangent and Cotangent Functions

Tangent and cotangent are also key trigonometric functions and are the ratios of sine and cosine. Tangent of an angle \( t \) equals the ratio \( \frac{sin(t)}{cos(t)} \) and cotangent is the reciprocal of this ratio, or \( \frac{cos(t)}{sin(t)} \). What's crucial to note is that when cosine equals zero, tangent is undefined because division by zero doesn't yield a real number.

In step 3 of our example, because \( cos(t) = 0 \), we correctly state that \( tan(t) \) is undefined. On the flip side, because \( sin(t) \) is nonzero, we can find the cotangent in step 6 by dividing zero by a nonzero number, which yields zero. The point to remember here is that whenever you encounter a zero in the denominator, the function value is undefined, as there is no number that can multiply by zero to give you anything besides zero.