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Chapter 13: Problem 4

Find the exact value of each trigonometric function. \(\sin 300^{\circ}\)

Short Answer

Expert verified
\(\sin 300^{\circ} = -\frac{\sqrt{3}}{2}\)

Step by step solution

01

Identify the reference angle

The angle given is \(300^{\circ}\). To find its reference angle, subtract it from \(360^{\circ}\). The reference angle \( \theta' \) is \(360^{\circ} - 300^{\circ} = 60^{\circ}\).
02

Determine the quadrant

The angle \(300^{\circ}\) is located in the fourth quadrant, as it lies between \(270^{\circ}\) and \(360^{\circ}\). In this quadrant, the sine function is negative.
03

Evaluate the sine of the reference angle

The reference angle we found is \(60^{\circ}\). The sine of \(60^{\circ}\) is \(\frac{\sqrt{3}}{2}\).
04

Apply the sign from the quadrant

Since \(300^{\circ}\) is in the fourth quadrant, where sine is negative, \(\sin 300^{\circ} = -\sin 60^{\circ} = -\frac{\sqrt{3}}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Reference Angles
Reference angles are crucial in trigonometry because they help us understand angles greater than 90 degrees more intuitively. When dealing with any angle, the reference angle is the smallest angle formed with the x-axis. It shares the same values of sine, cosine, and tangent as the original angle but may differ in sign depending on the quadrant. For example, with an angle like \(300^{\circ}\), you would find the reference angle by subtracting it from \(360^{\circ}\) (as it surpasses the \(270^{\circ}\) mark for the fourth quadrant). Thus, the reference angle is \(60^{\circ}\), which is the key to evaluating trigonometric functions for larger angles.
Exploring Quadrants in Trigonometry
The concept of quadrants is fundamental to trigonometry, dividing the plane into four distinct sections. These are known as:
  • First Quadrant: Both sine and cosine are positive.
  • Second Quadrant: Sine is positive, cosine is negative.
  • Third Quadrant: Both sine and cosine are negative.
  • Fourth Quadrant: Sine is negative, cosine is positive.

Knowing in which quadrant an angle lies helps determine the sign of its trigonometric functions. For instance, an angle like \(300^{\circ}\) falls in the fourth quadrant. Recognizing this quadrant-specific behavior ensures correct computation of angle-based problems.
Why the Sine Function is Negative in the Fourth Quadrant
In trigonometry, understanding why certain trigonometric values change signs in different quadrants is key. Sine, being a y-coordinate on the unit circle, turns negative in the fourth quadrant. This is because the y-values in this section of the coordinate plane are below the x-axis. While the reference angle \(60^{\circ}\) in this example has a positive sine value (\(\frac{\sqrt{3}}{2}\)), since \(300^{\circ}\) is in the fourth quadrant, the sine is adjusted to its negative counterpart: \(-\frac{\sqrt{3}}{2}\). Recognizing this pattern helps students confidently tackle similar trigonometric questions.

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Most popular questions from this chapter

Mr. Blackwell is building a triangular sandbox. He is to join a 3-meter beam to a 4 meter beam so the angle opposite the 4-meter beam measures \(80^{\circ} .\) To what length should Mr. Blackwell cut the third beam in order to form the triangular sandbox? Round to the nearest tenth.

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((5,12)\)

PREREQUISITE SKILL Find each value of \(\theta\) . Round to the nearest degree. $$ \cos \theta=-0.3420 $$

OPEN ENDED Give an example of a triangle that has two solutions by listing measures for \(A, a,\) and \(b,\) where a and \(b\) are in centimeters. Then draw both cases using a ruler and protractor.

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cot \left(\sin ^{-1} \frac{7}{9}\right)\)
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