Problem 11 - 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) \(0\)

Step 1: Understanding Negative Angles

The angle \(-\frac{\pi}{2}\) radians is a negative angle, which means it is measured in a clockwise direction from the positive x-axis. In the unit circle, this corresponds to rotating 90 degrees clockwise.

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

In the unit circle, the sine of an angle is given by the y-coordinate of the point where the terminal side of the angle intersects the circle. For the angle \(-\frac{\pi}{2}\), this point is (0, -1). Hence, \(\sin\left(-\frac{\pi}{2}\right) = -1\).

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

The cosine of an angle is given by the x-coordinate of the point where the terminal side of the angle intersects the unit circle. For \(-\frac{\pi}{2}\), this point is (0, -1), so \(\cos\left(-\frac{\pi}{2}\right) = 0\).

Step 4: Evaluate \(\cot\left(-\frac{\pi}{2}\right)\)

The cotangent of an angle is the reciprocal of the tangent, given by \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\). At \(-\frac{\pi}{2}\), \(\sin\left(-\frac{\pi}{2}\right) = -1\) and \(\cos\left(-\frac{\pi}{2}\right) = 0\). Since the cosine is 0, \(\cot\left(-\frac{\pi}{2}\right) = \frac{0}{-1} = 0\).

Key Concepts

unit circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. This makes it a powerful tool for understanding trigonometric functions. On the unit circle, every angle forms a corresponding point that can be described using the coordinates (x, y).

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

Using these coordinates, we can deduce values for sine and cosine. For example, to find \(\sin(-\frac{\pi}{2})\), we look at the y-coordinate of the point on the circle. On the unit circle, \(-\frac{\pi}{2}\) radians corresponds to the point \(0, -1\). Consequently, \(\sin(-\frac{\pi}{2}) = -1\). Understanding the unit circle makes it possible to determine trigonometric functions easily.

negative angles

In trigonometry, angles can be positive or negative. A positive angle is measured counterclockwise from the positive x-axis, while a negative angle is measured clockwise. Understanding negative angles allows us to extend trigonometric functions to cover all points on the circle.
For example, consider the angle \(\-\frac{\pi}{2}\). Rather than rotating counterclockwise, we rotate 90 degrees clockwise to locate the angle on the unit circle. This distinction between positive and negative angles is essential when identifying their corresponding points on the unit circle.

Recognizing whether an angle is positive or negative affects trigonometric calculations and can change the function values significantly. When working with trigonometric functions, always check the sign and direction of the angle to apply the correct method.

cotangent

Cotangent, noted as \(\cot\), is a trigonometric function that is the reciprocal of the tangent function. It is defined as follows:
\[\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\]
To calculate the cotangent, you must first determine the values of sine and cosine for the given angle. In the example of \(\-\frac{\pi}{2}\), \(\sin(-\frac{\pi}{2}) = -1\) and \(\cos(-\frac{\pi}{2}) = 0\). When you try to find \(\cot(-\frac{\pi}{2})\) using these values, you find:
\[\cot(-\frac{\pi}{2}) = \frac{0}{-1} = 0\]
When the cosine of an angle is 0, as it is here, the cotangent simplifies to 0. Understanding cotangent is key when interpreting different trigonometric problems, especially in cases involving angles that align with the axes of the unit circle.