Question

Find the exact value of each expression. Do not use a calculator. $$ \csc 300^{\circ} $$

Short answer

-\(\frac{2\sqrt{3}}{3}\)

Step-by-step solution

Understand the Reciprocal Identity

Firstly, recall the reciprocal identity of cosecant. The cosecant function is the reciprocal of the sine function. Thus, \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\)

Express the angle in terms of a common reference angle

Notice that \(\text{300}^{\text{\circ}}\) is an angle in the fourth quadrant. To simplify this, find the reference angle. The reference angle for 300° is \(\theta = 360^{\text{\circ}} - 300^{\text{\circ}} = 60^{\text{\circ}}\)

Determine the Sine of the Reference Angle

In the unit circle, \(\text{sin}(60^{\text{\circ}}) = \frac{\text{\sqrt{3}}}{2}\).

Adjust for the Quadrant

Because 300° is in the fourth quadrant, the sine is negative in this quadrant. Therefore \(\text{sin}(300^{\circ}) = -\frac{\sqrt{3}}{2}\).

Find the Cosecant Value

Finally, use the reciprocal identity: \(\text{csc}(300^{\circ}) = \frac{1}{\text{sin}(300^{\circ})} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}\).

Rationalize the Denominator

Multiply by \(\frac{\sqrt{3}}{\sqrt{3}}\) to rationalize: \(\text{csc}(300^{\circ}) = -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\).

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and their relationships. These functions relate the angles of a triangle to the lengths of its sides. The most common trigonometric functions include sine (sin), cosine (cos), and tangent (tan). Each function has a unique relationship with the angle and sides of a right triangle.
For instance, in a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse: \( \text {sin}(\theta) = \frac {\text {opposite}}{\text {hypotenuse}} \). Similarly, cosine is the ratio of the adjacent side to the hypotenuse: \( \text {cos}(\theta) = \frac {\text {adjacent}}{\text {hypotenuse}} \), and tangent is the ratio of the opposite side to the adjacent side: \( \text {tan}(\theta) = \frac {\text {opposite}}{\text {adjacent}} \).
These functions form the basis of various real-life applications, from architecture to signal processing.

Reciprocal Identities

Reciprocal identities are essential in trigonometry. Essentially, they express one trigonometric function as the reciprocal of another. The three main reciprocal identities are:

• Secant (sec), which is the reciprocal of cosine: \( \text {sec}(\theta) = \frac {1}{\text {cos}(\theta)} \).
• Cosecant (csc), which is the reciprocal of sine: \( \text {csc}(\theta) = \frac {1}{\text {sin}(\theta)} \). This identity was crucial in solving the given problem.
• Cotangent (cot), which is the reciprocal of tangent: \( \text {cot}(\theta) = \frac {1}{\text {tan}(\theta)} \).

Using these identities can simplify complex trigonometric expressions and assist in solving equations more efficiently.

Reference Angles

Reference angles help simplify the understanding of any angle by relating it to one of the acute angles (those between 0° and 90°). A reference angle is the smallest angle that the terminal side of any angle makes with the x-axis.
To find a reference angle:

• If the angle is in the first quadrant (0° to 90°), it is already the reference angle.
• For the second quadrant (90° to 180°), subtract the angle from 180°: \( \text {Reference Angle} = 180^\text {o} - (\text {given angle}) \).
• For the third quadrant (180° to 270°), subtract 180° from the angle: \( \text {Reference Angle} = (\text {given angle}) - 180^\text {o} \).
• For the fourth quadrant (270° to 360°), subtract the angle from 360°: \( \text {Reference Angle} = 360^\text {o} - (\text {given angle}) \). This was applied to the problem where the given angle was 300°.

Reference angles are useful because the trigonometric values of reference angles are the same for corresponding angles in other quadrants, with appropriate signs.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a valuable tool for understanding trigonometric functions and their values for various angles. Each point on the unit circle corresponds to an angle and the coordinates of the point give the values of cosine and sine for that angle.
The unit circle allows us to extend the definitions of sine, cosine, and tangent beyond the right triangle. Specifically:

• The x-coordinate of a point on the unit circle is \( \text {cos}(\theta) \).
• The y-coordinate is \( \text {sin}(\theta) \).
• Using these coordinates, we can determine other trigonometric values, such as tangent, secant, cosecant, and cotangent.

For example, for the angle 60°, the coordinates are \( \frac {1}{2} \) and \( \frac {\text {√3}}{2} \). This helps in calculating trigonometric values without a calculator.
Understanding the unit circle is crucial for solving many trigonometric problems, including the one provided, where we identified the sine value for 60° to find the cosecant of 300° due to the circle's symmetry.