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Chapter 6: Problem 14

Find the exact value of the trigonometric function. $$\sin 240^{\circ}$$

Short Answer

Expert verified
\( \sin 240^{\circ} = -\frac{\sqrt{3}}{2} \).

Step by step solution

01

Identify the reference angle

The angle 240掳 is in the third quadrant. To find the reference angle, subtract 180掳 from 240掳, giving us the reference angle of 60掳.
02

Determine the sine value using the reference angle

In the third quadrant, sine values are negative. Since the reference angle is 60掳, and knowing \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), the sine of 240掳 (using the negative sign) will thus be \( \sin 240^{\circ} = -\frac{\sqrt{3}}{2} \).
03

Confirm the angle position and sign

As 240掳 lies in the third quadrant and sine values are negative in this quadrant, the sign of our solution \( \sin 240^{\circ} = -\frac{\sqrt{3}}{2} \) is consistent with this rule.

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Key Concepts

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

Sine Function
The sine function is a fundamental concept in trigonometry that describes the relationship between angles and side lengths in a right-angled triangle. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse:\[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\]However, the sine function is not limited to just triangles; it is also part of the unit circle, a central concept in trigonometry. In the unit circle, the sine of an angle represents the y-coordinate of a point at that angle from the origin, with the radius of the circle being 1.
  • Sine is positive in the first and second quadrants.
  • Negative in the third and fourth quadrants.
  • The sine of 0掳 is 0, and it reaches a maximum of 1 at 90掳.
Reference Angle
The concept of a reference angle is crucial for understanding trigonometric functions beyond the first quadrant. A reference angle is the acute angle (less than 90掳) formed by the terminal side of an angle and the x-axis. No matter which quadrant the angle lies in, its reference angle will always help in determining the trigonometric function values. To find the reference angle, you usually follow these rules:
  • First Quadrant: The angle itself is the reference angle.
  • Second Quadrant: Subtract the angle from 180掳.
  • Third Quadrant: Subtract 180掳 from the angle.
  • Fourth Quadrant: Subtract the angle from 360掳.
This means for 240掳, which is in the third quadrant, the reference angle is 60掳, calculated as 240掳 鈭� 180掳.
Third Quadrant
Angles that lie in the third quadrant are those between 180掳 and 270掳. In this quadrant, it's important to remember that certain trigonometric functions will change their signs.
  • Sine and cosine are both negative in this quadrant.
  • The tangent function, however, is positive because both sine and cosine are negative, and dividing a negative by a negative yields a positive result.
Thus, if an angle like 240掳 is located here, its sine value, based on the reference angle and the quadrant rule, will be negative. In this case, since the sine of the reference angle 60掳 is \( \frac{\sqrt{3}}{2} \), the sine of 240掳 becomes \( -\frac{\sqrt{3}}{2} \). The negative sign is due to the positioning in the third quadrant.

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Most popular questions from this chapter

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