Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact value. (a) \(\csc 240^{\circ}\) (b) \(\csc \left(-330^{\circ}\right)\)

Step-by-Step Solution

Answer

(a) \(\csc 240^{\circ} = -\frac{2\sqrt{3}}{3}\); (b) \(\csc(-330^{\circ}) = -2\).

Step 1: Convert Degrees to Radians

The first step is to convert the angle from degrees to radians. We know that \[ 240^{\circ} = 240 \times \frac{\pi}{180} = \frac{4\pi}{3} \] and \[ -330^{\circ} = -330 \times \frac{\pi}{180} = -\frac{11\pi}{6} \]

Step 2: Find Reference Angle for 240°

The angle 240° is in the third quadrant. To find the reference angle, subtract 180° (since 180° is the lower bound of the third quadrant): \[ 240^{\circ} - 180^{\circ} = 60^{\circ} \]

Step 3: Find Cosecant for 240°

In the third quadrant, \sin \theta\ is negative, and the reference angle is 60°. The sine value for 60° is \sin 60^{\circ} = \frac{\sqrt{3}}{2}\, so \sin 240^{\circ} = -\frac{\sqrt{3}}{2}\. Thus, the cosecant is \[ \csc 240^{\circ} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \]

Step 4: Find Reference Angle for -330°

To convert -330° to a positive angle within a 0°-360° range, add 360°: \[ -330^{\circ} + 360^{\circ} = 30^{\circ} \]

Step 5: Find Cosecant for -330°

The reference angle for -330° is 30°. In the fourth quadrant, \sin \theta\ is negative. So, \sin 30^{\circ} = \frac{1}{2}\ leads to \sin(-330^{\circ}) = -\frac{1}{2}\. Thus, \[ \csc(-330^{\circ}) = -2 \]

Step 6: Combine Results

Consequently, the exact values are (a) \( \csc 240^{\circ} = -\frac{2\sqrt{3}}{3} \) (b) \( \csc(-330^{\circ}) = -2 \).

Key Concepts

Understanding Cosecant

Cosecant is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. In mathematical terms, \[ \csc(\theta) = \frac{1}{\sin(\theta)} \] This means that if you know the sine of an angle, you can easily find its cosecant by simply taking the reciprocal. Cosecant values come in handy, particularly when dealing with non-right triangles or periodic functions. It is essential to understand that cosecant is undefined wherever sine is zero, which happens at angles such as 0°, 180°, or 360°. Thus, it is vital to ensure the angle used has a non-zero sine value, to arrive at a valid result for cosecant.

Angle Conversion

Converting angles from degrees to radians is crucial in trigonometry since it is the standard unit of angle measurement used in calculus and higher mathematics. The conversion can be achieved using the formula:\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \] For example, for 240°, the conversion to radians is:\[ 240^{\circ} \times \frac{\pi}{180} = \frac{4\pi}{3} \] Similarly, angles can be expressed as positive to simplify calculations. When you encounter a negative angle, like -330°, it is often helpful to find its positive equivalent by adding 360°:\[ -330^{\circ} + 360^{\circ} = 30^{\circ} \] This makes working with angles much more straightforward, aligning them with standard circle positioning.

Reference Angle

A reference angle is the acute angle that a given angle makes with the x-axis. It reveals the simplest form of an angle that still preserves the angle's sine, cosine, or tangent properties but within the first quadrant. To determine the reference angle:

• For angles in the second quadrant, subtract from 180°.
• For angles in the third quadrant, subtract 180° from the angle.
• For angles in the fourth quadrant, subtract the angle from 360°.

In the case of 240°, which lies in the third quadrant, its reference angle is:\[ 240^{\circ} - 180^{\circ} = 60^{\circ} \] For -330°, after converting to 30°, the reference angle remains the same as it is already within the first quadrant.

Quadrants and Trigonometric Values

Understanding quadrants is fundamental in trigonometry because the sign of a trigonometric function changes based on the quadrant. The four quadrants on the Cartesian plane help determine whether sine, cosine, tangent, and their reciprocal functions are positive or negative:

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive, but cosine and tangent are negative.
• Third Quadrant: Tangent is positive, but sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, but sine and tangent are negative.

For instance, 240° is in the third quadrant; thus, the sine value is negative, resulting in a negative cosecant value. Similarly, -330° (or 30°) is in the fourth quadrant, leading to a negative sine and therefore a negative cosecant outcome.