Problem 15 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\csc \left(-630^{\circ}\right)$$

Step-by-Step Solution

Answer

The exact value of \( \csc(-630^{\circ}) \) is 1.

Step 1: Understand the Problem

We need to find the exact value of the cosecant function, \( \csc \theta \), given the angle \( -630^{\circ} \).

Step 2: Use Angle Reduction

To simplify the problem, we need to find an equivalent angle within the standard range \([0^{\circ}, 360^{\circ} )\). We do this by adding or subtracting multiples of \(360^{\circ}\) until the angle falls within this range. Compute \( -630^{\circ} + 720^{\circ} = 90^{\circ} \).

Step 3: Recall the Cosecant Function

The cosecant of an angle \( \theta \) is the reciprocal of the sine of that angle. Thus, \( \csc \theta = \frac{1}{\sin \theta} \).

Step 4: Find the Sine of the Angle

Now that we have the equivalent angle \( 90^{\circ} \), find \( \sin 90^{\circ} \). We know \( \sin 90^{\circ} = 1 \).

Step 5: Calculate the Cosecant Value

Using the reciprocal identity, calculate \( \csc 90^{\circ} = \frac{1}{\sin 90^{\circ}} = \frac{1}{1} = 1 \).

Key Concepts

Understanding Cosecant

The cosecant function, abbreviated as \( \csc \), is one of the six main trigonometric functions. It is defined as the reciprocal of the sine function. This means that for any given angle \(\theta\), \(\csc \theta = \frac{1}{\sin \theta}\). Understanding this relationship is crucial for solving problems involving cosecant.

• Cosecant does not exist for angles where the sine value is zero. This is because dividing by zero is undefined in mathematics.
• Common angles to consider are those where the sine value is either 1 or -1, such as \(90^{\circ}\) or \(270^{\circ}\).

Angle Reduction Simplified

Angle reduction is a technique used to simplify the computation of trigonometric functions for angles that are outside the standard range of \([0^{\circ}, 360^{\circ})\). By adjusting the angle by adding or subtracting full circles \((360^{\circ})\), the process brings the angle into a more manageable range.

• This method is especially useful in periodic functions like sine and cosine, as their values repeat every \(360^{\circ}\).
• For instance, with \(-630^{\circ}\), adding \(720^{\circ}\) (two full circles) results in \(90^{\circ}\).

Angle reduction simplifies computation and helps when visualizing the position of angles on the unit circle.

The Sine Function Explored

The sine function is fundamental to trigonometry and is represented as \( \sin \theta \). It outputs the y-coordinate of a point on the unit circle at a given angle \(\theta\) from the positive x-axis.

• On the unit circle, \( \sin \theta \) reaches its maximum value of 1 at \(90^{\circ}\) and its minimum of -1 at \(270^{\circ}\).
• The sine function is periodic with a period of \(360^{\circ}\), meaning it repeats its values every full circle.
• Understanding that \( \sin 90^{\circ} = 1 \) is critical, as it allows us to immediately know \( \csc 90^{\circ} \) without further calculation.

Recognizing these features enables seamless computation of trigonometric values in various mathematical scenarios.