Problem 37 - Algebra 2 and Trigonometry

Question

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \csc \theta=-\sqrt{2} $$

Step-by-Step Solution

Answer

The exact radian measure is \( \theta = -\frac{\pi}{4} \).

Step 1: Recall the definition of cosecant

The cosecant function is defined as the reciprocal of the sine function. Thus, \( \csc \theta = \frac{1}{\sin \theta} \). Given that \( \csc \theta = -\sqrt{2} \), we have \( \sin \theta = -\frac{1}{\sqrt{2}} \).

Step 2: Simplify the sine expression

The expression \( \sin \theta = -\frac{1}{\sqrt{2}} \) can be rationalized to \( \sin \theta = -\frac{\sqrt{2}}{2} \).

Step 3: Determine the reference angle

The value of \( \sin \theta = \pm \frac{\sqrt{2}}{2} \) corresponds to angles \( \theta = \frac{\pi}{4} \) (45 degrees) and its related angles in different quadrants.

Step 4: Consider sine's negative value

Since the sine is negative in the given equation, and sine is negative in the third and fourth quadrants, we consider angles in these quadrants: \( \theta = \pi + \frac{\pi}{4} \) and \( \theta = 2\pi - \frac{\pi}{4} \).

Step 5: Choose the angle with the smallest absolute value

Calculate the angle in radians: \( \theta = \frac{5\pi}{4} \) and \( \theta = \frac{7\pi}{4} \). The angle with the smallest absolute value is \( \theta = -\frac{\pi}{4} \), which lies in the fourth quadrant and has an equivalent positive angle of \( \frac{7\pi}{4} \).

Key Concepts

Cosecant Function

The cosecant function is an essential concept in trigonometry. It is closely related to the sine function, and when we talk about cosecant, we are delving into the idea of mathematical reciprocals. The relationship is defined as:

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

This means that the cosecant of an angle \( \theta \) is the reciprocal of the sine of that angle. Therefore, understanding the sine function helps us grasp the cosecant function more effectively.

For example, if \( \sin \theta = 0.5 \), then \( \csc \theta = \frac{1}{0.5} = 2 \). This reciprocal relationship is pivotal when analyzing trigonometric equations and is used to solve equations where the sine value is given.

Sine Function

The sine function is one of the fundamental trigonometric functions. It helps in understanding angles and their ratios in right-angled triangles. The sine of an angle \( \theta \) can be represented geometrically as the ratio of the length of the side opposite to \( \theta \) to the hypotenuse in a right triangle. It is often expressed as:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

But beyond geometric interpretations, the sine function is crucial in periodic functions, describing the oscillations and waves. Its values range between -1 and 1, and it follows a repeating cycle as the angle increases. This periodicity helps in predicting patterns and behaviors in mathematical and real-world applications. Recognizing when sine values are positive or negative is essential for solving trigonometric problems, such as determining the quadrants where specific angles lie.

Reference Angles

Reference angles are a simplified way to handle trigonometric problems. It is the smallest angle that an angle makes with the x-axis, helping in finding equivalent angles in different quadrants.

• A reference angle is always between \( 0 \) and \( \frac{\pi}{2} \) radians (or \( 0 \text{ to } 90 \) degrees).
• It allows us to easily calculate trigonometric functions of any angle by using known values within the first quadrant.

To find a reference angle for a given angle \( \theta \), we consider its position in the unit circle, adjusting to its equivalent angle within the first quadrant. This concept is notably helpful in simplifying the evaluation of trig functions and understanding the symmetrical properties of the unit circle.

Quadrants in Trigonometry

Quadrants play a crucial role in interpreting the signs and values of trigonometric functions. The unit circle is divided into four quadrants based on the Cartesian coordinate plane were angles are measured.

• **First Quadrant**: Both sine and cosine values are positive.
• **Second Quadrant**: Sine is positive while cosine is negative.
• **Third Quadrant**: Both sine and cosine are negative.
• **Fourth Quadrant**: Sine is negative while cosine is positive.

Each angle corresponding to these quadrants gives distinct trigonometric values. The quadrant determines the sign of the sine, cosine, and other trigonometric functions. Understanding which quadrant an angle resides in simplifies the solving and understanding of trigonometric equations, like the one discussed in the exercise, where we identify which quadrants have negative sine values.