Problem 29 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \csc 270^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \csc 270^{\circ} \) is \(-1\).

Step 1: Understanding Cosecant

Cosecant is the reciprocal of the sine function. Therefore, \( \csc \theta = \frac{1}{\sin \theta} \). To find the value of \( \csc 270^{\circ} \), we first need to determine \( \sin 270^{\circ} \).

Step 2: Determine Sine of 270 Degrees

The angle \( 270^{\circ} \) is located on the negative y-axis in the unit circle. In this position, the sine, which corresponds to the y-coordinate on the unit circle, is \( -1 \). Therefore, \( \sin 270^{\circ} = -1 \).

Step 3: Calculate Cosecant of 270 Degrees

Using the relationship \( \csc 270^{\circ} = \frac{1}{\sin 270^{\circ}} \), substitute \( \sin 270^{\circ} = -1 \) to get \( \csc 270^{\circ} = \frac{1}{-1} = -1 \).

Key Concepts

trigonometric functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are fundamental in studying geometry, physics, and engineering. The primary trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each of these functions has a corresponding reciprocal function:

• Cosecant (\( \csc \theta \)) is the reciprocal of sine.
• Secant (\( \sec \theta \)) is the reciprocal of cosine.
• Cotangent (\( \cot \theta \)) is the reciprocal of tangent.

These functions help us understand the behavior of angles and arcs in the context of circles and triangles. One key objective when using trigonometric functions is solving for unknown angles or sides in right triangles. These functions also extend to modeling periodic phenomena, like sound waves and tides, due to their cyclic nature.

unit circle

The unit circle is a powerful tool in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle allows us to understand the relationships between angles and the sine and cosine functions.Here's how it works:

• Every angle around the unit circle can be represented in radians or degrees. An angle's corresponding point on the circle has coordinates (\( \cos \theta, \sin \theta \)).
• The angle 270° lies on the negative y-axis on the unit circle. At this point, the coordinates are (0, -1), indicating that \( \cos 270^{\circ} = 0 \) and \( \sin 270^{\circ} = -1 \).

Using the unit circle, it becomes straightforward to evaluate trigonometric functions for common angles like 0°, 90°, 180°, and 270°. This simplicity makes it an essential visualization tool for solving trigonometric problems and aiding in the comprehension of more complex trigonometric identities and equations.

reciprocal trigonometric functions

Reciprocal trigonometric functions serve as the inverse counterparts to the primary trigonometric functions and are equally important in mathematical contexts. They include:

• Cosecant (\( \csc \theta \)), defined as \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant (\( \sec \theta \)), defined as \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent (\( \cot \theta \)), defined as \( \cot \theta = \frac{1}{\tan \theta} \).

These functions are used to find values that traditional direct trigonometric functions cannot easily provide, especially in the context of certain angles.For example, to find cosecant for the angle 270°, you first need to know \( \sin 270^{\circ} = -1 \). Substituting this value, \( \csc 270^{\circ} = \frac{1}{\sin 270^{\circ}} = \frac{1}{-1} = -1 \). Understanding reciprocal trigonometric functions allows for deeper insight into the geometrical relationships on the unit circle and enhances the ability to solve more complex trigonometric equations.