Problem 118 - Precalculus

Question

Use a calculator to evaluate the following expressions. If you get an error, explain why. $$\csc \left(-270^{\circ}\right)$$

Step-by-Step Solution

Answer

\(\csc(-270^\circ) = 1\)

Step 1: Understand the Function

The given expression is the cosecant of \(-270^\circ\). The cosecant function, \(\csc(x)\), is the reciprocal of the sine function, i.e., \(\csc(x) = \frac{1}{\sin(x)}\). So, we first need to find \(\sin(-270^\circ)\).

Step 2: Find the Sine of the Angle

The angle \(-270^\circ\) is a full rotation of \(-360^\circ\) minus \(90^\circ\). Therefore, it's equivalent to \(90^\circ\) in the standard position, but in the clockwise direction. The sine of \(90^\circ\) is \(1\), so \(\sin(-270^\circ) = 1\).

Step 3: Calculate Cosecant

Now that we know \(\sin(-270^\circ) = 1\), we can calculate the cosecant: \(\csc(-270^\circ) = \frac{1}{1} = 1\).

Key Concepts

Cosecant

The cosecant function is an important concept in trigonometry and is often represented as \(\csc(x)\). It is defined as the reciprocal of the sine function. In simple terms, this means that cosecant is the inverse of sine. More specifically, if you know the sine of an angle, you can easily determine the cosecant by performing this calculation: \(\csc(x) = \frac{1}{\sin(x)}\). It is vital to note that for the cosecant to be defined, the sine value (\(\sin(x)\)) must not be zero. This is because division by zero is undefined in mathematics.

When calculating the cosecant of an angle, you first need to determine the sine value for that particular angle. Once you have the sine value, simply take its reciprocal to find the cosecant. Remembering this relationship is crucial for solving problems that involve reciprocal trigonometric functions.

Sine Function

The sine function is one of the fundamental trigonometric functions and is commonly written as \( \sin(x) \). It is crucial in the study of triangles, particularly right-angled triangles, where it represents the ratio of the length of the side opposite the angle to the hypotenuse.

In the context of the unit circle, the sine of an angle is defined as the y-coordinate of the point on the circle at that angle, extending from the circle's center at the origin \( (0,0) \). The sine function is periodic with a period of \(360^{\circ}\) or \(2 \pi\) radians, meaning it repeats its values every full circle.

Common angles and their sine values are useful to remember:

• \( \sin(0^{\circ}) = 0 \)
• \( \sin(90^{\circ}) = 1\)
• \( \sin(180^{\circ}) = 0 \)
• \( \sin(270^{\circ}) = -1 \)
• \( \sin(360^{\circ}) = 0 \)

The sine function is also an odd function, which mathematically means that \( \sin(-x) = -\sin(x) \). This property can be helpful when working with negative angles, as seen with \( \sin(-270^{\circ}) = 1 \).

Angle Measurement

Understanding how angles are measured is foundational in trigonometry. Angles are typically measured in degrees or radians. Degrees are the most common and come from dividing a full circle into 360 parts. Radians, on the other hand, are based on the radius of a circle and provide a direct relationship between the angle's arc length and the radius.

An angle is considered positive when measured counterclockwise from the positive x-axis and negative when measured clockwise. In our example, \(-270^{\circ}\) is negative, implying a clockwise measurement from the positive x-axis to the terminal side of the angle. It's equivalent position is \(90^{\circ}\) in the standard position because \(-270^{\circ}\) represents a rotation of \(-360^{\circ} + 90^{\circ}\).

Some key angle equivalences to remember include:

• \(0^{\circ}\) is the same as \(360^{\circ}\).
• \(90^{\circ}\) is a quarter of a full circle turn.
• \(180^{\circ}\) represents a half-circle turn.
• \(270^{\circ}\) is three-quarters of a full circle.

By normalizing angles within \(0^{\circ}\) to \(360^{\circ}\) via these equivalencies, it becomes easier to visualize their positions and understand their trigonometric relationships.