Problem 17 - Algebra and Trigonometry

Question

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: Recognize the Function and Angle

We need to find the exact value of the cosecant function for the angle \(-630^\circ\). Cosecant is the reciprocal of sine, so \(\csc\theta = \frac{1}{\sin\theta}\). The angle \-630^\circ\ is negative, which indicates it is measured in the clockwise direction.

Step 2: Find a Positive Coterminal Angle

To find a coterminal angle in the standard position, add \(360^\circ\) until a positive angle is obtained. Calculate \-630^\circ + 720^\circ = 90^\circ\. Therefore, \(-630^\circ\) is coterminal with \(90^\circ\).

Step 3: Evaluate the Sine of the Positive Angle

Determine the sine of the coterminal angle. \(\sin 90^\circ = 1\).

Step 4: Determine the Cosecant Value

The cosecant is the reciprocal of sine, hence \(\csc 90^\circ = \frac{1}{1} = 1\). Since \(-630^\circ\) is coterminal with \(90^\circ\), \(\csc(-630^\circ) = \csc(90^\circ) = 1\).

Key Concepts

Cosecant

The cosecant function is one of the six fundamental trigonometric functions that relate to right triangles and the unit circle. It is denoted as \( \csc \theta \) and is the reciprocal of the sine function. That means:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This function helps us understand ratios in right triangles and can be found using the unit circle. Where sine represents the ratio of the opposite side over the hypotenuse in a right triangle, cosecant represents the ratio of the hypotenuse over the opposite side.

It's important to note that the sine function cannot be zero for cosecant to exist because division by zero is undefined. Consequently, the cosecant of angles like \( \theta = 0 \degree \) and \( \theta = 180 \degree \) doesn't exist since their sine is zero. Learning cosecant offers insights into wave patterns, oscillations, and harmonic functions, as it provides useful reciprocals in mathematical analysis.

Coterminal Angles

Coterminal angles are angles in the standard position that share the same terminal side. They differ by a full rotation, which means adding or subtracting multiples of \( 360\degree \) (or \( 2\pi \) radians) to find them. In the context of this exercise, you often need to find coterminal angles to simplify the computation of trigonometric functions.

• An angle \( \theta \) has coterminal angles at \( \theta + 360n \degree \), where \( n \) is an integer.
• For negative angles, add \( 360\degree \) repeatedly until you find the corresponding positive coterminal angle.

For example, the angle \( -630\degree \) is coterminal with \( 90\degree \). By adding \( 720\degree \) (or two full circles) to \( -630\degree \), we find an equivalent angle that is much simpler to work with. Coterminality aids us in understanding how angles relate in various applications, such as rotations and periodic functions.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are essential members of the trigonometric family and include cosecant, secant, and cotangent. They are directly related to the more familiar sine, cosine, and tangent functions as follows:

• Cosecant (\( \csc \theta \)) is the reciprocal of sine \( (\sin \theta ) \)
• Secant (\( \sec \theta \)) is the reciprocal of cosine \( (\cos \theta ) \)
• Cotangent (\( \cot \theta \)) is the reciprocal of tangent \( (\tan \theta ) \)

These functions are critical for solving more complex trigonometric equations and understanding relationships in wave mechanics and harmonic motion. They are particularly useful in contexts where the regular trigonometric functions equal zero or are undefined.

In this exercise, we focused on the cosecant, using its reciprocal nature to simplify expressions and find explicit function values. This approach helps handle trigonometric expressions elegantly, providing insights into angle behaviors and periodic properties, often used in higher-level mathematics.