Problem 14 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \left(-\frac{3 \pi}{2}\right) \quad\) (b) \(\cos \left(-\frac{3 \pi}{2}\right) \quad\) (c) \(\cot \left(-\frac{3 \pi}{2}\right)\)

Step-by-Step Solution

Answer

(a) 1, (b) 0, (c) 0

Step 1: Understand the Angle

The problem asks for trigonometric functions at the angle \(-\frac{3\pi}{2}\). Since this is a negative angle, let's convert it to a positive angle. Angles that differ by full rotations of \(2\pi\) have identical trigonometric values, so we add \(2\pi\) to \(-\frac{3\pi}{2}\) to get the equivalent positive angle. \(-\frac{3\pi}{2} + 2\pi = \frac{\pi}{2}\).

Step 2: Evaluate \( \sin(\frac{\pi}{2}) \)

Knowing that \(\frac{\pi}{2}\) is \(90^\circ\), the value of \(\sin(\frac{\pi}{2})\) is \(1\) as it is the same as the sine of \(90^\circ\). Thus, \(\sin(-\frac{3\pi}{2}) = \sin(\frac{\pi}{2}) = 1\).

Step 3: Evaluate \( \cos(\frac{\pi}{2}) \)

At \(\frac{\pi}{2}\) or \(90^\circ\), the value of \(\cos(\frac{\pi}{2})\) is \(0\), same as the cosine of \(90^\circ\). Therefore, \(\cos(-\frac{3\pi}{2}) = \cos(\frac{\pi}{2}) = 0\).

Step 4: Calculate \( \cot(\frac{\pi}{2}) \)

The cotangent function is defined as \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\). For \(\theta = \frac{\pi}{2}\), we have \(\cos(\frac{\pi}{2}) = 0\) and \(\sin(\frac{\pi}{2}) = 1\). Therefore, \(\cot(\frac{\pi}{2}) = \frac{0}{1} = 0\). Thus, \(\cot(-\frac{3\pi}{2}) = 0\).

Key Concepts

Sine Function

The sine function is a fundamental part of trigonometry. It helps us understand the relationship between angles and side lengths in right-angled triangles. The sine of an angle, denoted as \( \sin(\theta) \), is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. This means if you have a triangle with an angle \( \theta \), then:

• Opposite side = the side directly opposite to angle \( \theta \)
• Hypotenuse = the longest side in the triangle, opposite the right angle

So, the sine function is: \[ \sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \]Trigonometric functions like sine are periodic, which means their values repeat at regular intervals. For sine, this interval is \( 2\pi \). Thus, angles that differ by a full rotation (or multiple rotations) of \( 2\pi \) have the same sine values. For instance, \( \sin(\frac{\pi}{2}) = 1 \), indicating that the height of the sine wave at this angle is maximum.

Cosine Function

The cosine function is another essential trigonometric function closely related to the sine function. In a right triangle, the cosine of an angle \( \theta \), symbolized as \( \cos(\theta) \), compares the length of the adjacent side to the hypotenuse. When you look at a triangle, here's what you see:

• Adjacent side = the side next to angle \( \theta \) that is not the hypotenuse
• Hypotenuse = the longest side opposite the right angle

The cosine function formula is:\[ \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \]Like the sine function, cosine is periodic with a period of \( 2\pi \). This repetitive nature ensures that adding \( 2\pi \) to any angle does not change its cosine value. For example, \( \cos(\frac{\pi}{2}) = 0 \), meaning the cosine wave crosses the x-axis at this point.

Cotangent Function

The cotangent function is a bit less common than sine and cosine, but it is equally important. It is the reciprocal of the tangent function. The cotangent of an angle \( \theta \), expressed as \( \cot(\theta) \), is the ratio of the cosine to the sine of that angle. Here's the definition:\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]Cotangent compares the lengths of sides as:

• Adjacent side = the side next to the angle
• Opposite side = the side facing the angle

The key takeaway is that if the sine of an angle is zero (like at \( \theta = \frac{\pi}{2} \)), the cotangent will also be zero because dividing by a non-zero cosine gives zero. However, if the cosine is zero, this poses an undefined scenario, but not at \( \theta = \frac{\pi}{2} \). Hence, at \( \frac{\pi}{2} \), \( \cot(\frac{\pi}{2}) = 0 \), explaining our solution.