Problem 35 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \csc 245^{\circ} $$

Step-by-Step Solution

Answer

\(\csc 245^{\circ} \approx -1.1036\)

Step 1: Understand the Cosecant Function

The cosecant function, denoted as \(\csc\), is the reciprocal of the sine function. For an angle \(\theta\), \(\csc \theta = \frac{1}{\sin \theta}\). Our task is to find \(\csc 245^{\circ}\).

Step 2: Compute \(\sin 245^{\circ}\)

Find the sine of \(245^{\circ}\) using a calculator. Ensure the calculator is set to degree mode. Calculating gives \(\sin 245^{\circ} \approx -0.9063\).

Step 3: Calculate \(\csc 245^{\circ}\)

Now that you have \(\sin 245^{\circ}\), find \(\csc 245^{\circ}\) by taking the reciprocal: \(\csc 245^{\circ} = \frac{1}{\sin 245^{\circ}} = \frac{1}{-0.9063} \approx -1.1036\).

Step 4: Final Answer

The calculated value of \(\csc 245^{\circ}\) is approximately \(-1.1036\). Ensure the answer is rounded to four decimal places as required.

Key Concepts

Understanding Cosecant

The cosecant function is a fundamental part of trigonometry and is symbolized by \( \csc \). It forms one of the six main trigonometric functions. Cosecant is especially important because it helps us in analyzing angles and their relationships in right-angled triangles.

• The cosecant of an angle is simply the reciprocal of the sine function. In mathematical terms, this is written as \( \csc \theta = \frac{1}{\sin \theta} \). This means that if you know the sine of an angle, you can easily find the cosecant by taking the reciprocal.
• In any scenario where you encounter the sine value, you can instantly understand that the cosecant is just the 'flipped' or inverse version.
• Knowing this reciprocal identity helps streamline the calculation process for various trigonometric problems.

When dealing with angles beyond acute or simple ones, such as the example \( 245^{\circ} \), recognizing that these functions can be obtained using a calculator simplifies the process immensely.

The Role of the Sine Function

The sine function, noted as \( \sin \), is a principal function in trigonometry that measures the ratio between the opposite side and the hypotenuse in a right-angled triangle.

• This function is crucial because it lays the groundwork for understanding angles and distances in various fields such as physics and engineering.
• In terms of a circle, \( \sin \theta \) represents the y-coordinate of the point that results from rotating a point on the unit circle by an angle \( \theta \).

For an angle like \( 245^{\circ} \), this can seem tricky since it’s not a standard angle like \( 30^{\circ} \) or \( 90^{\circ} \), but modern calculators allow us to compute values like \( \sin 245^{\circ} \) quickly. For example, by setting your calculator to degree mode, you can find that \( \sin 245^{\circ} \approx -0.9063 \). It's the basis for finding the cosecant and other trigonometric identities.

Introducing Reciprocal Identities

Reciprocal identities are a group of equalities in trigonometry that show how functions such as sine, cosine, and tangent relate to their respective reciprocals: cosecant, secant, and cotangent.

• Cosecant is the reciprocal of sine, as shown by the identity \( \csc \theta = \frac{1}{\sin \theta} \).
• These identities allow for deeper analysis and calculations within trigonometry by simplifying complex expressions into simpler ones using known values.
• Understanding and using reciprocal identities are fundamental skills for anyone working with advanced trigonometry.

In our example, once we find \( \sin 245^{\circ} \approx -0.9063 \), determining \( \csc 245^{\circ} \) becomes straightforward through the reciprocal identity: \( \csc 245^{\circ} = \frac{1}{-0.9063} \approx -1.1036 \). Such calculations highlight the practical utility of reciprocal identities in problem-solving.