High School Trigonometry/Measuring Rotation

In this lesson you will learn about angles of rotation, which are found in many different real phenomena. Consider, for example, a game that is played with a spinner. When you spin the spinner, how far has it gone?

You can answer this question in several ways. You could say something like "the spinner spun around three times". This means that the spinner made three complete rotations, and then landed back where it started.

We can also measure the rotation in degrees. In the previous lesson we worked with angles in triangles, measured in degrees. You may recall from geometry that a full rotation is 360 degrees, usually written as 360°. Half a rotation is then 180° and a quarter rotation is 90°. Each of these measurements will be important in this lesson, as well as in the remainder of the chapter.

Learning Objectives

 * Determine if an angle is acute, right, obtuse, or straight.
 * Express the measure of angles in degrees, minutes, and seconds.
 * Express the measure of angles in decimal degrees.
 * Identify and draw angles of rotation in standard position.
 * Identify quadrantal angles.
 * Identify co-terminal angles.

Acute, Right, Obtuse, and Straight Angles
In general, angles are categorized by their size. The table below summarizes the categories, which might be familiar from the previous lesson.


 * {| class="wikitable" style="width:520px;"

! Name || Description
 * + Table 1.8
 * Acute || An angle whose measure is less than 90 degrees.
 * Right || An angle whose measure is exactly 90 degrees.
 * Obtuse || An angle whose measure is more than 90 degrees, but less than 180 degrees.
 * Straight || An angle whose measure is exactly 180 degrees.
 * }
 * Obtuse || An angle whose measure is more than 90 degrees, but less than 180 degrees.
 * Straight || An angle whose measure is exactly 180 degrees.
 * }
 * }

You should make sure that you can visually determine which category an angle belongs to.

Determine if the angle is acute, right, obtuse, or straight.

a.
 * Measuring Rotation Example 1a.svg

b.
 * Measuring Rotation Example 1b.svg

c.
 * Measuring Rotation Example 1c.svg

Solution:

a. This angle is an acute angle.

If it is difficult to categorize the angle visually, you can compare it to a right angle. Doing this will help you see that the angle is smaller than a right angle.


 * Measuring Rotation Solution 1a.svg

b. This angle is an obtuse angle.

Again, you can compare the angle to a right angle, if needed.

c. This angle is a right angle.

It is important to note that usually a right angle is marked with a small square.


 * Measuring Rotation Solution 1c.svg

It is also important to note that you can determine the measure of an angle using a protractor. This measure will of course be an approximation, as no protractor is perfect and the person measuring cannot perfectly line up the protractor or hold it steady.

Use a protractor to measure the angle in example 1a.

Solution:

The angle is about 50°.


 * Measuring Rotation Solution 2.png

When working with angles measured in degrees, we often report our answers using a decimal, such as 78.5°. However, in some contexts, angles are measured using fractional parts.

Measuring Angles
Two wheels are in direct contact. The radius of one is .5 meters. The radius of the other is 1 meter. The smaller one rotates four full turns. How many rotations does the larger wheel make? How many degrees does the larger wheel rotate through?

Solution:

Every time the small wheel rotates once, its entire circumference passes along the larger wheel, C = 2π(.5). Since the circumference of the large wheel is 2π(1), the large wheel rotates half way around. So if the small wheel rotates 4 times, or 360 · 4 = 1440° the large wheel rotates 2 times, or 360 · 2 = 720°.

We can measure angles in much the same way we measure time. A minute is $$\tfrac{1}{60}$$ of a degree. A second is $$\tfrac{1}{60}$$ of a minute, so it is $$\tfrac{1}{360}$$ of a degree. For example, 48°20′45″ is the way we write 48 degrees, 2 minutes, and 45 seconds. We can write this angle using fraction notation, as well as decimal notation:


 * $$48^\circ 20' 45'' = 48 + \frac{20}{60} + \frac{45}{360} = 48 + \frac{1}{3} + \frac{1}{8} = 48\frac{11}{24} = 48.458\overline{3} \approx 48.4583^\circ$$

We can also write a decimal degree using degrees, minutes, and seconds. For example, we can rewrite 125.88° if we write the decimal part as a fraction:


 * $$0.88 = \frac{88}{100} = \frac{x}{60}$$

Now solve for x:




 * style="text-align:right; padding-bottom:10px;" | $$\frac{88}{100} =$$
 * style="padding-bottom:10px;" | $$\frac{x}{60}$$
 * style="text-align:right; padding-bottom:10px;" | $$\frac{44}{50} =$$
 * style="padding-bottom:10px;" | $$\frac{x}{60}$$
 * style="text-align:right; padding-bottom:10px;" | $$44 \times 60 =\,\!$$
 * style="padding-bottom:10px;" | $$50x\,\!$$
 * style="text-align:right; padding-bottom:10px;" | $$2640 =\,\!$$
 * style="padding-bottom:10px;" | $$50x\,\!$$
 * style="text-align:right;" | $$x =\,\!$$
 * $$52.8\,\!$$
 * }
 * style="text-align:right;" | $$x =\,\!$$
 * $$52.8\,\!$$
 * }
 * }

Now we have 125.88° = 125°52.8′. We need to write .8 minutes as seconds:




 * style="text-align:right; padding-bottom:10px;" | $$\frac{0.8}{60} =$$
 * style="padding-bottom:10px;" | $$\frac{s}{360}$$
 * style="text-align:right; padding-bottom:10px;" | $$0.8 \cdot 360 =$$
 * style="padding-bottom:10px;" | $$60x\,\!$$
 * style="text-align:right; padding-bottom:10px;" | $$288 =\,\!$$
 * style="padding-bottom:10px;" | $$60s\,\!$$
 * style="text-align:right; padding-bottom:10px;" | $$s =\,\!$$
 * style="padding-bottom:10px;" | $$\frac{288}{60}$$
 * style="text-align:right;" | $$s =\,\!$$
 * $$4.8\,\!$$
 * }
 * style="text-align:right;" | $$s =\,\!$$
 * $$4.8\,\!$$
 * }
 * }

Therefore 125.88° = 125°52′4.8″.

Notice that the angle 125.88° is an obtuse angle. Its measure is less than 180°. What does an angle look like that is more than 180°? More than 360°?

Next you will learn about a particular way to represent angles that will allow you to represent 180°, 360°, or any other angle.

Angles of Rotation in Standard Position
We can use our knowledge of graphing to represent any angle. The figure below shows an angle in what is called standard position.


 * Standard position.svg

The initial side of an angle in standard position is always on the positive x−axis. The terminal side always meets the initial side at the origin. Notice that the rotation goes in a counterclockwise direction. This means that if we rotate clockwise, we are generating a negative angle. Below are several examples of angles in standard position.


 * {| style="width:600px;"


 * style="padding-right:20px;" | 45 degree standard position.svg || 90 degree standard position.svg
 * style="padding-right:20px; padding-top:20px;" | 225 degree standard position.svg || style="padding-top:20px;" | -45 degree standard position.svg
 * }
 * }

The 90 degree angle is one of four quadrantal angles. A quadrantal angle is one whose terminal side lies on an axis. Along with 90°, 0°, 180° and 270° are quadrantal angles.


 * {| style="width:600px;"


 * style="padding-right:20px;" | 0 degree angle.svg || 90 degree angle.svg
 * style="padding-right:20px; padding-top:20px;" | 180 degree angle.svg || style="padding-top:20px;" | 270 degree angle.svg
 * }
 * }

These angles are referred to as quadrantal because each angle defines a quadrant. Notice that without the arrow indicating the rotation, 270° looks as if it is a 90°, defining the fourth quadrant. Notice also that 360° would look just like 0°. The difference is in the action of rotation. This idea of two angles actually being the same angle is discussed next.

Coterminal Angles
Consider the angle 30°, in standard position.


 * 30 degree standard position.svg

Now consider the angle 390°. We can think of this angle as a full rotation (360°), plus an additional 30 degrees.


 * 390 degree angle.svg

Notice that 390° looks the same as 30°. Formally, we say that the angles share the same terminal side. Therefore we call the angles co-terminal. Not only are these two angles co-terminal, but there are infinitely many angles that are co-terminal with these two angles. For example, if we rotate another 360°, we get the angle 750°. Or, if we create the angle in the negative direction (clockwise), we get the angle −330. Because we can rotate in either direction, and we can rotate as many times as we want, we can keep generating angles that are co-terminal with 30°.

Which angles are co-terminal with 45°?

a. -45°

b. 405°

c. -315°

d. 135°

Solution:

b. 405° and c. -315° are co-terminal with 45°.


 * 45, -315, and 405 co-terminal angles.svg

Notice that terminal side of the first angle, −45°, is in the 4th quadrant. The last angle, 135° is in the 2nd quadrant. Therefore neither angle is co-terminal with 45°.

Now consider 405°. This is a full rotation, plus an additional 45 degrees. So this angle is co-terminal with 45°. The angle −315° can be generated by rotating clockwise. To determine where the terminal side is, it can be helpful to use quadrantal angles as markers. For example, if you rotate clockwise 90 degrees 3 times (for a total of 270 degrees), the terminal side of the angle is on the positive y−axis. For a total clockwise rotation of 315 degrees, we have 315 − 270 = 45 degrees more to rotate. This puts the terminal side of the angle at the same position as 45°.

Lesson Summary
In this lesson we have categorized angles according to their size, and we have extended our knowledge of angles to include angles of rotation. We have defined what it means for an angle to be in standard position, and we have looked at angles in standard position, including the quadrantal angles. We have also defined the concept of co-terminal angles. All of the ideas in this lesson will be used in the following lesson, to define the trigonometric functions that are the focus of this chapter.

Points to Consider

 * How can one angle look exactly the same as another angle?
 * Where might you see angles of rotation in real life?

Review Questions
(a)
 * 1) Determine if the angle is acute, right, obtuse, or straight.

(b)


 * 1) Approximate the measure of the angle.  Explain how you approximated.
 * 1) Rewrite the measure of each angle in degrees, minutes, and seconds.
 * (a) 85.5°
 * (b) 12.15°
 * (c) 114.96°
 * 1) Rewrite the measure of each angle in decimal degrees.
 * (a) 54°10'25"
 * (b) 17°40'5"
 * 1) Determine the measure of the angle between the clock hands at the given time.
 * (a) 6:00
 * (b) 3:00
 * (c) 1:00
 * 1) Through what angle does the minute hand of a clock rotate between 12:00am and 1am?
 * 2) A car goes around a 90 degree circular curve in a racetrack. The diameter of an automobile's wheel is .6 m. The distance between the wheels is 2 m. The radius of the curve the car is following is 100 m measured at the closest wheel to the track. What is the difference in number of rotations that the outer wheel must turn compared with the inner wheel?
 * 3) State the measure of an angle that is co-terminal with 90°.
 * 4) Name two angles that are co-terminal with 120°.
 * (a) An angle that is negative.
 * (b) An angle that is greater than 360°.
 * 1) A drag racer goes around a 180 degree circular curve in a racetrack in a path of radius 120 m. Its front and back wheels have different diameters. The front wheels are .6 m in diameter. The rear wheels are much larger; they have a diameter of 1.8 m. The axles of both wheels are 2 m long. Which wheel has more rotations going around the curve. How many more degrees does that wheel rotate compared with the wheel that rotates the least making that curve?

Review Answers

 * (a) Acute
 * (b) Straight
 * 1) The angle is about 120 degrees. You can approximate the measure of the angle using a protractor, or by using other angles, such as 90 and 30.
 * (a) 85°30'
 * (b) 12°9'
 * (c) 114°57'36"
 * (a) ≈ 54.236°
 * (b) ≈ 17.681°
 * (a) 180°
 * (b) 90°
 * (c) 30°
 * 1) 360°
 * 2) 5/6
 * 3) Answers will vary.  Examples: 450°, -270°.
 * 4) Answers will vary.  Examples: -240°, 480°.
 * 5) The front wheel rotates more. It rotates 100 revolutions versus 33.89 revolutions for the back wheel, which is a ∼ 23800 degree difference.
 * 1) 5/6
 * 2) Answers will vary.  Examples: 450°, -270°.
 * 3) Answers will vary.  Examples: -240°, 480°.
 * 4) The front wheel rotates more. It rotates 100 revolutions versus 33.89 revolutions for the back wheel, which is a ∼ 23800 degree difference.

Vocabulary

 * acute angle
 * An acute angle is an angle with measure between 0 and 90 degrees.


 * co-terminal angles
 * Angles of rotation in standard position are co-terminal of they share the same terminal side.


 * minutes
 * A minute is $$\tfrac{1}{60}$$ of a degree.


 * obtuse angle
 * An obtuse angle is an angle with measure between 90 and 180 degrees.


 * protractor
 * A protractor is a tool used to measure angles.


 * quadrantal angle
 * A quadrantal angle is an angle in standard position whose terminal side lies on an axis.


 * right angle
 * A right angle is an angle with measure exactly 90 degrees.


 * seconds
 * A second is $$\tfrac{1}{60}$$ of a minute, or $$\tfrac{1}{3600}$$ of a degree.


 * standard position
 * An angle in standard position has its initial side on the positive x−axis, its vertex at the origin, and its terminal side anywhere in the plane. A positive angle means a counterclockwise rotation. A negative angle means a clockwise rotation.


 * straight angle
 * A straight angle is an angle with measure 180 degrees. A straight angle makes a straight line.