Problem 25 - Precalculus: A Concise Course

Question

Evaluate (if possible) the sine, cosine, and tangent of the real number. $$ t=-\frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

The sine of -3π/2 equals -1, the cosine equals 0, but the tangent is undefined since it implies division by zero.

Step 1: Calculate Sine

First, to compute the sine of the given angle (-3π/2), substitute t into the sine function, yielding: \[sin(-3π/2)\]. Since the sine of -3π/2 equals -1, \(sin(-3π/2) = -1\)

Step 2: Calculate Cosine

Next, compute the cosine of the given angle by substituting the value of t into the cosine function: \[cos(-3π/2)\]. Since the cosine of -3π/2 equals 0, hence \(cos(-3π/2) = 0\)

Step 3: Calculate Tangent

Finally, to compute the tangent of the given angle, substitute t into the tangent function: \[tan(-3π/2)\]. The tangent of an angle is given by the ratio of the sine to the cosine of the angle. However, since the cosine of -3π/2 is 0, the tangent is undefined as division by zero is undefined: \[tan(-3π/2) = \frac{sin(-3π/2)}{cos(-3π/2)} = \frac{-1}{0}\] which is undefined.

Key Concepts

Evaluate Sine Cosine Tangent

Understanding how to evaluate trigonometric functions is a cornerstone of geometry and pre-calculus. The sine, cosine, and tangent functions help us describe the relationship between the angles and sides of a triangle, and they are fundamental in exploring circular motion and harmonic patterns.

Let's take the given problem as an example where we have to evaluate sine, cosine, and tangent for the angle \( t = -\frac{3 \pi}{2} \). Keep in mind that the trigonometric functions for specific angles can often be determined using the unit circle or reference angles.

• Sine (\(\sin\)): It represents the ratio of the opposite side to the hypotenuse in a right triangle. At the angle \( -\frac{3 \pi}{2} \), the sine value is \(\sin(-\frac{3 \pi}{2}) = -1\).
• Cosine (\(\cos\)): This ratio compares the adjacent side to the hypotenuse of a right triangle. The cosine of \( -\frac{3 \pi}{2} \) is exactly \(\cos(-\frac{3 \pi}{2}) = 0\).
• Tangent (\(\tan\)): Tangent compares the opposite side to the adjacent side, and it equals the sine divided by the cosine of the angle. For this angle, we find an undefined value because it involves division by zero: \(\tan(-\frac{3 \pi}{2}) = \frac{\sin(-\frac{3 \pi}{2})}{\cos(-\frac{3 \pi}{2})} = \frac{-1}{0}\), which is not a valid mathematical operation.

Remember, these values relate to the coordinates on the unit circle at specific angles, reflecting the function's behavior in a full 360° or \(2\pi\) radians rotation.

Trigonometric Ratios

Trigonometric ratios are essential for understanding the properties of angles and their measurement. They relate angles to the sides of right triangles and are integral in various applications including physics, engineering, and astronomy.

There are six main trigonometric ratios: sine, cosine, and tangent (covered here), along with their reciprocals, cosecant, secant, and cotangent. Each of these functions provides a unique way to represent the ratio of sides within a right triangle or the circular function of a particular angle.

The key is to remember their definitions:

• Sine (\(\sin\)) - opposite/hypotenuse
• Cosine (\(\cos\)) - adjacent/hypotenuse
• Tangent (\(\tan\)) - opposite/adjacent

Memorizing these ratios is less daunting when you imagine them as coordinates of a circle with a radius of one, known as the unit circle. As the angle varies, these ratios change, resulting in different values of trigonometric functions.

Undefined Tangent

In trigonometry, not all functions will yield a real number for every angle. Specifically, tangent can become undefined at certain angles. This occurs when the denominator of the tangent ratio, which is the cosine, is zero. Since division by zero is not possible in real numbers, the tangent function is undefined at angles where cosine equals zero.

In our example with \( t = -\frac{3 \pi}{2} \), tangent is undefined because the cosine value is zero. Graphically, think of tangent as the slope of the line created by the angle in the unit circle. When the line goes straight up or down, it becomes a vertical line with an undefined slope.

Remember that the tangent function will be undefined at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) (or odd multiples of \( \frac{\pi}{2} \) radians), corresponding to 90° and 270°, where the function attempts to divide by zero. Recognizing these angles quickly can save time and confusion when solving trigonometric problems.