Problem 210 - Precalculus

Question

For the following exercises, use reference angles to evaluate the expression. $$ \csc 150^{\circ} $$

Step-by-Step Solution

Answer

\( \csc 150^{\circ} = 2 \)

Step 1: Understanding the Reference Angle

First, we need to find the reference angle for the given angle. The reference angle is the smallest angle to the x-axis, which can be found by subtracting the given angle from 180° if the angle is in the second quadrant. For \( 150^{\circ} \), it is in the second quadrant, so the reference angle is \( 180^{\circ} - 150^{\circ} = 30^{\circ} \).

Step 2: Understanding the Cosecant Function

The cosecant function \( \csc \theta \) is the reciprocal of the sine function, so \( \csc \theta = \frac{1}{\sin \theta} \). We are interested in finding \( \csc 150^{\circ} \), which will relate to finding \( \csc \) of the reference angle \( 30^{\circ} \).

Step 3: Calculating \( \sin 30^{\circ} \)

We need to determine the sine of the reference angle \( 30^{\circ} \). From basic trigonometric identities, we know that \( \sin 30^{\circ} = \frac{1}{2} \).

Step 4: Calculating \( \csc 150^{\circ} \)

Since \( 150^{\circ} \) shares its reference angle with \( 30^{\circ} \) and cosecant is positive in the second quadrant, \( \csc 150^{\circ} = \frac{1}{\sin 150^{\circ}} = \frac{1}{\sin 30^{\circ}} \). Substitute the value from Step 3: \( \csc 150^{\circ} = \frac{1}{\frac{1}{2}} = 2 \).

Key Concepts

Reference Angles

In trigonometry, a reference angle is a helpful tool used to find the equivalent angle from the x-axis to the terminal side of any given angle.
Reference angles allow us to simplify the problems involving angles greater than 90 degrees. Here's why reference angles are important:

• They are generally positive and less than or equal to 90°.
• They help in determining the trigonometric function values for larger angles by using known values for these small angles.

Understanding how to determine a reference angle is crucial. For example, if an angle is located in the second quadrant, as is the case with 150°, the reference angle can be calculated by subtracting the given angle from 180°.
That's because in standard position, angles are measured counter-clockwise from the positive x-axis.
The reference angle for 150° is calculated as follows:
\[ 180° - 150° = 30° \]The result indicates that 150° has a reference angle of 30°.
This small angle is then used to evaluate trigonometric functions.

Cosecant Function

The cosecant function is defined as the reciprocal of the sine function. If you know the sine of an angle, you can easily find its cosecant by using the relationship:
\[ \csc \theta = \frac{1}{\sin \theta} \]This relationship highlights that whenever sine is known, obtaining cosecant becomes straightforward.
While learning about the cosecant function, remember:

• It is undefined whenever the sine of an angle is zero, as you cannot divide by zero.
• Its value is more than or equal to one, or less than or equal to negative one, since sine won’t exceed the value of 1.

Let's apply this to the problem of finding \( \csc 150° \).
First, examine \( \sin 150° \) which is equal to \( \sin 30° \) because 150° shares a reference angle of 30° in the second quadrant.
From basic trigonometric knowledge, we know:
\[ \sin 30° = \frac{1}{2} \]Therefore, the cosecant of 150°, given that the sine of 150° equals the sine of 30°, is:\[ \csc 150° = \frac{1}{\sin 150°} = \frac{1}{\frac{1}{2}} = 2 \]

Trigonometric Identities

Trigonometric identities are equations that relate various trigonometric functions to one another. They are crucial for solving trigonometry problems involving any angle.
Some common trigonometric identities include:

• Pythagorean identities, such as: \[ \sin^2 \theta + \cos^2 \theta = 1 \]
• Reciprocal identities like: \[ \csc \theta = \frac{1}{\sin \theta}, \text{and similarly for cosecant, secant, and cotangent.} \]
• Angle sum and difference identities.

When evaluating trigonometric functions, identities can simplify and provide alternative ways to calculate expressions.
In our exercise, the reciprocal identity helped us find the value of \( \csc 150° \).
By calculating \( \csc \theta \) using the sine value, it simplified the process and led us to the correct result efficiently.