Problem 13 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Angle

The angle given is \(-\frac{\pi}{2}\). This is equivalent to \(-90°\). We need to understand where this angle lies on the unit circle. It is in the fourth quadrant, where the angle goes clockwise from the positive x-axis.

Step 2: Evaluating \(\sin(-\frac{\pi}{2})\)

The sine of an angle corresponds to the y-coordinate of the point on the unit circle. At \(-\frac{\pi}{2}\) or \(-90°\), the point on the unit circle is (0, -1). Thus, \(\sin(-\frac{\pi}{2}) = -1\).

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

The cosine of an angle corresponds to the x-coordinate of the point on the unit circle. At \(-\frac{\pi}{2}\) or \(-90°\), the point on the unit circle is (0, -1). Thus, \(\cos(-\frac{\pi}{2}) = 0\).

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

The cotangent of an angle is the reciprocal of the tangent, \(\cot(\theta) = \frac{1}{\tan(\theta)}\), and \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Since \(\cos(-\frac{\pi}{2}) = 0\), the tangent is undefined, and thus \(\cot(-\frac{\pi}{2}) = \frac{\cos(-\frac{\pi}{2})}{\sin(-\frac{\pi}{2})} = \frac{0}{-1} = 0\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential concepts in mathematics. They deal with the ratios of the sides of right triangles and the angles within them. In a unit circle, a circle with a radius of 1, these functions help us understand positions and angles.
Key trigonometric functions include:

• Sine (\( ext{sin}\)): Measures the ratio of the opposite side to the hypotenuse.
• Cosine (\( ext{cos}\)): Measures the ratio of the adjacent side to the hypotenuse.
• Tangent (\( ext{tan}\)): Expressed as the ratio of sine to cosine.
• Cotangent (\( ext{cot}\)): The reciprocal of tangent.

Using the unit circle, trigonometric functions can determine angle measures and coordinate points corresponding to those angles. Understanding these concepts forms the foundation of trigonometry.

Sine and Cosine

Sine and cosine are two of the primary trigonometric functions, both defined using the unit circle. The sine of an angle is associated with the y-coordinate, whereas the cosine is associated with the x-coordinate.
In the unit circle:

• Sine represents the vertical position of a point moving around the circle.
• Cosine represents the horizontal position of a point along the circle.

For example, at the angle \(-\frac{\pi}{2}\) (or \(-90^\circ\)), the sine is \(-1\), as the point is directly below the origin, while the cosine is \(0\), indicating that there is no horizontal displacement.Understanding these values is crucial for solving trigonometric problems and analyzing wave patterns.

Cotangent

Cotangent is a lesser-known, yet equally important, trigonometric function. It is defined as the reciprocal of the tangent function. This means it is the ratio of the adjacent side to the opposite side.The formula for cotangent is:\[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\]An important aspect to be aware of is that cotangent becomes undefined when tangent is zero. This occurs when cosine is zero or the angle is \(-\frac{\pi}{2}\).In such cases, it results in a division by zero, which is mathematically undefined. Thus, at \(-\frac{\pi}{2}\), cotangent equals zero. Recognizing these conditions is vital when working with cotangent in various mathematical contexts.