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Chapter 6: Problem 9

9鈥�32 Find the exact value of the trigonometric function. $$\sin 150^{\circ}$$

Short Answer

Expert verified
\( \sin 150^{\circ} = \frac{1}{2} \)

Step by step solution

01

Recognize the Reference Angle

First, recognize that the given angle is 150掳. Since this angle is in the second quadrant, find the reference angle. The reference angle for an angle greater than 90掳 and less than 180掳 can be found by subtracting the given angle from 180掳. Thus, the reference angle for 150掳 is 180掳 - 150掳 = 30掳.
02

Determine the Sine Value of the Reference Angle

Recall the sine value of the reference angle. The sine of 30掳 is a well-known trigonometric value. From trigonometric tables or knowledge, we know that \( \sin 30^{\circ} = \frac{1}{2} \).
03

Determine the Sign of Sin in the Second Quadrant

In the unit circle, the sine function is positive in the first and second quadrants. Since 150掳 is in the second quadrant, \( \sin 150^{\circ} \) will have the same positive sign as \( \sin 30^{\circ} \).
04

Conclude with the Sine Value

Since the reference angle 30掳 has a sine value of \( \frac{1}{2} \) and the sine function is positive in the second quadrant, \( \sin 150^{\circ} = \frac{1}{2} \).

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Key Concepts

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

Reference Angle
When working with angles, especially in trigonometry, the concept of a reference angle is extremely helpful. A reference angle is the smallest angle that can be formed between the terminal side of a given angle and the x-axis.
It is always a non-negative angle ranging between 0掳 and 90掳. Identifying the reference angle allows you to simplify the calculation of trigonometric functions.
  • For angles in the second quadrant (like 150掳), the reference angle is found by subtracting the given angle from 180掳: thus, for 150掳, the reference angle is 180掳 - 150掳 = 30掳.
  • In the third quadrant, subtract the angle from 180掳 again, while in the fourth quadrant, subtract the angle from 360掳.
Knowing the reference angle helps to determine the trigonometric function values for larger angles.
Unit Circle
The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Understanding the unit circle can help to visualize angles and calculate trigonometric functions efficiently.
  • The angle's position on the unit circle is defined in counterclockwise from the positive x-axis.
  • The circle's quadrants determine the sine (y-coordinate) and cosine (x-coordinate) values' positivity or negativity.
For example, when an angle like 150掳 is plotted on the unit circle, it lies in the second quadrant, where sine values are positive, making it easy to decide the sign of the sine function.
Sine Function
The sine function is one of the primary trigonometric functions, and it is expressed as the y-coordinate of a point on the unit circle for a given angle. This function tells us how far "up or down" a point on the circle is.
  • The sine of an angle is the length of the side opposite the angle, divided by the hypotenuse in a right triangle perspective.
  • The sine function is periodic, repeating every 360掳 or 2蟺 radians.
Specifically, for the angle 150掳, finding oindentewlineoindent oindentewlineoindent the sine value means first finding its reference angle (30掳) and knowing from the unit circle or trigonometric tables that oindentewlineoindent oindentewlineoindent oindentewlineoindent oindentewlineoindent \( \sin 30^{\circ} = \frac{1}{2} \).This value is reused for 150掳 but adjusted for the correct sign based on its location in the unit circle.
Quadrants
Each of the four sections that the coordinate plane is divided into is called a quadrant. Understanding in which quadrant an angle falls can tell you the signs of the trigonometric functions.
  • Quadrant I: positive sine, cosine, and tangent.
  • Quadrant II: positive sine and negative cosine and tangent.
  • Quadrant III: negative sine, cosine, and positive tangent.
  • Quadrant IV: negative sine and positive cosine and tangent.
For the given problem, 150掳 lies in the second quadrant. This information is crucial because it means that the sine function for 150掳 will have a positive value, just like the sine of its reference angle, 30掳.
Exact Values
Exact values in trigonometry provide precise, non-rounded values for trigonometric functions at key angles. These values are rooted in the special angles, such as 30掳, 45掳, 60掳, and they typically appear as simple fractions or square roots that are well-known and used frequently.
  • The sine of 30掳 is an important exact value: \( \sin 30^{\circ} = \frac{1}{2} \).
  • For angles like 150掳, which is in the second quadrant, the sine takes the same positive value as its reference angle, 30掳.
Having these exact values memorized or easily accessible can dramatically speed up solving trigonometric problems and understanding relationships between different angles.

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Most popular questions from this chapter

In Exercise 72 of Section 6.1 a method was given for finding the radius of the earth. Here is a more modern method: From a satellite 600 mi above the earth, it is observed that the angle formed by the vertical and the line of sight to the horizon is \(60.276^{\circ} .\) Use this information to find the radius of the earth.

To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle \(75^{\circ}\) from the horizontal. An observer 600 m away measures the angle of elevation to the spot of light to be \(45^{\circ}.\) Find the height \(h\) of the cloud cover.

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Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=26, \quad c=15, \quad \angle C=29^{\circ} $$

Clocks and Angles In one hour, the minute hand on a clock moves through a complete circle, and the hour hand moves through \(\frac{1}{12}\) of a circle. Through how many radians do the minute and the hour hand move between \(1 : 00 \mathrm{P.M.}\) and \(6 : 45\) P.M. (on the same day)?
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