Problem 14 - Algebra 2

Question

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \csc 60^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \csc 60^{\circ} \) is \( 2 / \sqrt{3} \)

Step 1: Define the trigonometric function

Cosecant (\( \csc \)) is defined as the reciprocal of sine and is therefore not defined at angles where sine equals 0. However, for this exercise, the angle 60 degrees is used where the sine value is not 0.

Step 2: Find the sine value

Consulting the unit circle or a trigonometric value table, the sine of 60 degrees, written as \( \sin 60^{\circ} \), is equal to \( \sqrt{3} / 2 \).

Step 3: Calculate the cosecant value

The cosecant is found by taking the reciprocal value of sine. Hence, \( \csc 60^{\circ} = 1 / ( \sqrt{3} / 2 ) = 2 / \sqrt{3} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, cycling through values as an angle progresses around a circle. They relate the angles of a triangle to the lengths of its sides.

• Sine (\( \sin \)): This function represents the opposite side divided by the hypotenuse in a right triangle.
• Cosecant (\( \csc \)): The cosecant is the reciprocal of sine. It is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).
• Other basic functions: Include cosine (\( \cos \)) and tangent (\( \tan \)). Each has its reciprocal: secant (\( \sec \)) and cotangent (\( \cot \)).

Understanding these functions is pivotal as they are used in calculating distances, waves, and angles, playing crucial roles in both geometry and algebra.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to a specific angle and provides a geometric interpretation of trigonometric functions.

• Angles and coordinates: On the unit circle, an angle’s terminal side intersects the circle at a specific point. The coordinates of this point \((x, y)\) represent the cosine and sine of the angle: \( x = \cos \theta \), \( y = \sin \theta \).
• Special angles: Certain angles, like \( 30^{\circ} \), \( 45^{\circ} \), and \( 60^{\circ} \), have well-known sine, cosine, and tangent values. For \( 60^{\circ} \), \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \) and \( \cos 60^{\circ} = \frac{1}{2} \).

Using the unit circle allows for a straightforward visualization of trigonometric functions and their properties when angles change.

Reciprocal Identities

Reciprocal identities are used to express one trigonometric function in terms of another. These are particularly helpful when dealing with functions like cosecant, secant, and cotangent, which are not typically covered in basic angles. They provide a method to switch between functions using reciprocals.

• Cosecant: \( \csc \theta = \frac{1}{\sin \theta} \), based on the identification that cosecant is the inverse of sine.
• Secant: Expressed as \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent: Essentially the reciprocal of tangent, written as \( \cot \theta = \frac{1}{\tan \theta} \).

For example, in the problem of finding \( \csc 60^{\circ} \), we recognize that it is the reciprocal of \( \sin 60^{\circ} \), calculated as \( \frac{2}{\sqrt{3}} \). This articulation simplifies relationships between functions and enhances problem-solving versatility.