Problem 17 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The value of \( \csc (-630°) \) is 1.

Step 1: Determine Coterminal Angle

To find the coterminal angle of -630°, we add multiples of 360° because the trigonometric functions have a period of 360°. Calculate -630° + 360° * 2 = 90°. So, the coterminal angle is 90°.

Step 2: Use Basic Trigonometric Identity

Recall the definition of cosecant: \(\csc\theta = \frac{1}{\sin\theta}.\)So, \(\csc 90° = \frac{1}{\sin 90°}.\)

Step 3: Find Sine of the Angle

Sine of 90° is a known value: \(\sin 90° = 1.\)

Step 4: Calculate Cosecant

Substitute the value of \(\sin 90° = 1\) into the cosecant formula: \(\csc 90° = \frac{1}{1} = 1.\)

Key Concepts

Understanding Coterminal Angles

A coterminal angle is an angle that shares the same initial and terminal sides with another angle. They are essentially angles that end up in the same position on the unit circle. To find coterminal angles, you can add or subtract multiples of 360° (or 2π radians if you're working with radians). This is because a full circle in standard position spans 360°, so adding or subtracting that cycle will bring you back to the same spot.
In the given problem, to find the coterminal angle of -630°, we add multiples of 360°.

• First, calculate \(-630° + 360° \times 2\) which results in 90°.
• Thus, the coterminal angle of -630° is 90°.

Using coterminal angles is helpful because it allows us to simplify the function evaluation by converting it into a more manageable angle value.

Cosecant Function Basics

The cosecant function, denoted as \(\csc(\theta)\), is the reciprocal of the sine function. In simple terms, this means that the cosecant of an angle is the inverse of its sine. Therefore, \(\csc(\theta) = \frac{1}{\sin(\theta)}\). This indicates that for values where the sine function is zero, the cosecant is undefined, as division by zero is not possible in mathematics.
In the solution, after finding that the coterminal angle is 90°, we identify that \(\csc(90°)\) needs evaluation.

• According to the definition, \(\csc(90°) = \frac{1}{\sin(90°)}\).
• This example demonstrates how the cosecant function depends directly on the value of the sine function.

Understanding this relationship is crucial when evaluating trigonometric functions as it shows how knowing one function can help determine values of another.

The Sine Function and Its Importance

The sine function, represented as \(\sin(\theta)\), is a fundamental trigonometric function that describes the y-coordinate of a point on the unit circle for a given angle \(\theta\). It is one of the basic building blocks for understanding how angles interact within a circle.
For the sine of 90°, it’s important to remember that

• When an angle is 90°, the terminal side falls on the positive y-axis of the unit circle.
• This means the sine of 90° is the y-coordinate of this point, which is 1.

So, \(\sin(90°) = 1\).
Recognizing these values, especially those at key angles like 0°, 90°, 180°, and 270°, is extremely helpful when solving trigonometric equations, as these often come up in many problems and solutions.