Lesson 6.1 Intro to Angles

Standard Position

An angle with its vertex at the origin and its initial side along the positive x-axis

Coterminal Angles

Angles in standard position with the same terminal side.

360°= ? π radians 180°= ? π radians

360°= 2π radians 180°= π radians

Terminal Side

A ray of an angle that rotates about the center

Negative Angles Measured in Degrees

sorry, there were no pictures, but see the reference for positive angles measured in degrees, and move the measures from each quadrant clockwise. (-Quad I= Quad 4, -Quad II= Quad III, -Quad III= Quad II, -Quad IV= Quad 1)

Negative Angles Measured in Radians

sorry, there were no pictures, but see the reference for positive angles measured in radians, and move the measures from each quadrant clockwise. (-Quad I= Quad 4, -Quad II= Quad III, -Quad III= Quad II, -Quad IV= Quad 1)

Which of the following statements is not true concerning angle​ measure? Select the correct choice below. A.If an angle has negative​ measure, then the direction of its rotation is clockwise. B.The angle in standard position formed by rotating the terminal side of angle one complete counterclockwise rotation has a radian measure of π radians. C.If an angle has positive​ measure, then the direction of its rotation is counterclockwise. D.The angle in standard position formed by rotating the terminal side of an angle one complete counterclockwise rotation has a measure of 360 degrees

B.The angle in standard position formed by rotating the terminal side of angle one complete counterclockwise rotation has a radian measure of π radians.

Initial Side

The fixed ray of an angle

Which of the following statements best describes an angle that is in standard​ position? Select the correct choice below. A.An angle is in standard position if the vertex is at the origin of a rectangular coordinate system and the initial side lies along the positive​ x-axis. B.An angle is in standard position if the vertex is at the origin of a rectangular coordinate system and the initial side lies along the negative​ y-axis. C.An angle is in standard position if the vertex is at the origin of a rectangular coordinate system and the initial side lies along the negative ​ x-axis. D.An angle is in standard position if the vertex is at the origin of a rectangular coordinate system and the initial side lies along the positive​ y-axis.

A.An angle is in standard position if the vertex is at the origin of a rectangular coordinate system and the initial side lies along the positive​ x-axis.

Which of the following statements best describes two coterminal​ angles? Select the correct answer below. A.Two angles in standard position are coterminal if they have the same terminal side. B.Two angles in standard position are coterminal if they both have the same amount of rotation. C.Two angles in standard position are coterminal if they have the same initial side. D.Two angles in standard position are coterminal if the terminal sides of both angles lie in the same quadrant.

A.Two angles in standard position are coterminal if they have the same terminal side.

Which of the following statements is true concerning the conversion between degree and radian​ measure? Select the correct choice below. A.To convert from degrees to​ radians, multiply by 180 degrees and divide by π. B.To convert from radians to​ degrees, multiply by 180 degrees and divide by π. C.To convert from radians to​ degrees, multiply by π and divide by 180 degrees. D.To convert from radians to​ degrees, multiply by 2π and divide by 360 degrees

B.To convert from radians to​ degrees, multiply by 180 degrees and divide by π.

Which of the following statements is not true concerning radian​ measure? Select the correct choice below. A.One radian is the measure of a central angle that has an intercepted arc equal in length to the radius of the circle. B.An angle in standard position having a radian measure of θ=−5π/3 has a terminal side that lies in Quadrant I. C.An angle in standard position having a radian measure of θ=−11π/6 has a terminal side that lies in Quadrant IV. D.One radian is approximately 57.3°.

C.An angle in standard position having a radian measure of θ=−11π/6 has a terminal side that lies in Quadrant IV.

Finding Coterminal Angles Using Radian Measure

Coterminal angles can be obtained by adding any nonzero integer multiple of 2π a given angle.

Finding Coterminal Angles using Degree Measure

Coterminal angles can be obtained by adding any nonzero integer multiple of 360° to a given angle.

Least Non-Negative Measure

Every angle has a coterminal angle of least non-negative measure. If θ is a given angle, then we use the notation θC to denote the angle of least non-negative measure coterminal with θ.

Using Radian Measure to Find the Least Non-Negative Measure of Coterminal Angles Ex. Find the angle of least nonnegative​ measure, θC​, that is coterminal with θ=−37π/9.

Recall that coterminal angles are angles in standard position having the same terminal side. An angle of least nonnegative​ measure, θC​, is the smallest angle greater than or equal to 0 that is coterminal with the given angle. Coterminal angles can be obtained by adding a multiple of 2π. Recall that given any angle θ and any nonzero integer​ k, the angles θ and θ+k•2π are coterminal angles.​ Therefore, choose values for k and evaluate to find a coterminal angle. Since the given angle θ is​ negative, add positive values of k to get to the angle with least nonnegative measure coterminal with θ. To find the angle with least nonnegative measure that is coterminal to −37π/9 and less than 2π​, add three multiples of 2π. Thus k=3. Substitute for θ and k and evaluate the expression θ+k•2π. −37π/9 +​ (3)(2π​) =−37π/9 + 6π =−37π/9 + 54π/9 =17π/9 An angle that measures −37π/9 radians is coterminal with an angle that measures 17π/9 radians. ​Therefore, θC= 17π/9.

Using Radian Measure to Find the Least Non-Negative/Negative Measures of Coterminal Angles Within A Given Range Ex. Find the angle of least nonnegative measure that is coterminal with −5π/3. Then find the measure of the negative angle that is coterminal with −5π/3 such that the angle lies between −4π and −2π.

Recall that coterminal angles are angles in standard position having the same terminal side. An angle of least nonnegative​ measure, θC​, is the smallest angle greater than or equal to 0 that is coterminal with the given angle. Coterminal angles can be obtained by adding any nonzero integer multiple of 2π to a given angle. The angles coterminal with θ=−5π/3 have the form −5π/3+k•2π​, where k is an integer. Since θ is a negative​ angle, choose positive values of k until the angle has a measure that is greater than or equal to 0. Let k=​1, evaluate θ+k•2π. −5π/3+​(1)•2π =−5π/3+2π =−5π/3+6π/3 =π/3 Since the new angle is now greater than or equal to​ 0, the least nonnegative angle coterminal with −5π/3 is π/3. A negative coterminal angle is found by adding a negative multiple of 2π. In this​ case, the negative angle that is between −4π and −2π can be obtained by one more clockwise rotation. Let k=−​1, evaluate θ+k•2π. −5π/3+​(−​1)•2π =−5π/3−2π =−5π/3−6π/3 =−11π/3 The negative angle that is coterminal to −5π/3 and between−4π and −2π is −11π/3.

Using Degree Measure to Find the Least Non-Negative Measure of Coterminal Angles Ex. Find the angle of least nonnegative​ measure, θC​, that is coterminal with θ = −910°

Recall that coterminal angles are angles in standard position having the same terminal side. An angle of least nonnegative​ measure, θC​, is the smallest angle greater than or equal to​ 0° that is coterminal with the given angle. Coterminal angles can be obtained by adding any nonzero integer multiple of 360° to a given angle. The angles coterminal with θ=−910° have the form −910°+​k(360°​) where k is an integer. Since θ is a negative​ angle, choose positive values of k until the angle of least nonnegative measure is found. Let k=​1, evaluate −910°+​k(360°​) = −910°+​(1)(360°​) = −910°+360° =−550° So −550° is a negative angle that is coterminal with −910°. ​However, since −550° is still a negative​ angle, there is another value of k that will make the new coterminal angle greater than or equal to 0°. Let k=​2, evaluate −910°+​k(360°​). −910°+​(2)(360°​) =−910°+720° =−190° So −190° is another angle that is coterminal with −910° and is still negative. Continue trying values of k until an angle with the least nonnegative measure that is coterminal with −910° has been found. Let k=​3, evaluate −910°+​k(360°​). −910°+​(3)(360°​) =−910°+1080° =170° When k is​ 3, the measure of the new angle is nonnegative.​ Therefore, the least nonnegative angle that is coterminal with −910° has measure 170°. ​Therefore, θC is 170°.

Using Degree Measure to Find the Least Non-Negative/Negative Measures of Coterminal Angles Within A Given Range Ex. Find the angle of least nonnegative measure that is coterminal with −8°. Then find the measure of the negative angle that is coterminal with −8° such that the angle lies between −720° and −360°.

Recall that coterminal angles are angles in standard position having the same terminal side. An angle of least nonnegative​ measure, θc​, is the smallest angle greater than or equal to​ 0° that is coterminal with the given angle. ​Remember, coterminal angles can be found by adding a multiple of 360°. Recall that given any angle θ and any nonzero integer​ k, the angles θ and θ+k•360° are coterminal angles. These multiples of 360° may be positive or negative. Since the angle measuring −8° and is between −360° and 0°​, adding 360° produces the least nonnegative coterminal angle. The least nonnegative coterminal angle is −8°+360°=352°. A negative coterminal angle is found by adding a negative multiple of 360. In this​ case, the negative angle that is between −720° and −360° can be obtained by one more clockwise​ rotation, k=−1. −8°+(−1)•360° −8°+​(−360°​)=−368° The negative angle that is coterminal to −8° and between−720° and −360° is −368°.

Using Radian Measure to Find two positive/negative Coterminal Angles Within A Given Range Ex. Find two positive and two negative angles that are coterminal with the quadrantal angle θ=−2π such that each angle lies between −6π to 4π.

Recall that given any angle θ and any nonzero integer​ k, the angles θ and θ+k•2π are coterminal angles. ​Therefore, choose values for k and evaluate to find a coterminal angle. When k=​1, the expression θ+k•2π has been evaluated below.​ Remember, θ=−2π. −2π+​(1)•2π =−2π+2π =0 An angle of measure 0 is neither positive nor negative.​ Thus, it is not one of the desired angles even though it is in the correct range. When k=−​1, the expression θ+k•2π has been evaluated below.​ Remember, θ=−2π. −2π+​(−​1)•2π =−2π+​(−2π​) =−4π Since −4π falls between −6π and 4π​, then −4π is another angle that is coterminal with −2π. Two angles have been found 0 and −4π. Continue trying values of k. Recall that the angle found must be between −6π and 4π. When k=​2, the expression θ+k•2π will yield the​ following: −2π+​(2)•2π =−2π+4π =2π When k=−​2, the expression θ+k•2π will yield the​ following: −2π+​(−​2)•2π =−2π+​(−4π​) =−6π When k=​3, the expression θ+k•2π will yield the​ following: −2π+​(3)•2π =−2π+​(6π​) =4π The coterminal angles found from the previous steps are −6π​, −4π​, 2π​, and 4π. ​ Thus, the two positive angles between −6π and 4π are 2π and 4π. The two negative angles between −6π and 4π are −6π and −4π.

Using Degree Measure to Find two positive/negative Coterminal Angles Within A Given Range Ex. Find two positive and two negative angles that are coterminal with the quadrantal angle θ=−810° such that each angle lies between −1080° and 720°.

Recall that given any angle θ and any nonzero integer​ k, the angles θ and θ+k•360° are coterminal angles. ​Therefore, choose values for k and evaluate to find a coterminal angle. In this​ case, θ=−810°. Evaluate the expression θ+k•360° when k=1. −810°+​(1)•360° =−810°+360° =−450° Since −450° falls between −1080° and 720°​, −450° is one of the desired coterminal angles. Continue to find other coterminal angles by trying other values of k. Evaluate the expression θ+k•360° when k=2. ​Remember, θ=−810°. −810°+​(2)•360° =−810°+720° =−90° Since −90° falls between −1080° and 720°​, −90° is one of the desired coterminal angles. Evaluate the expression θ+k•360° when k=3. −810°+​(3)•360° =−810°+1080° =270° Evaluate the expression θ+k•360° when k=4. −810°+​(4)•360° =−810°+1440° =630° The coterminal angles found above are −450°​, −90°​, 270°​, and 630°. Thus, the two positive angles between −1080° and 720° are 270° and 630°. The two negative angles between −1080° and 720° are −90° and −450°.

Converting Radians to Degrees Ex. Convert the given angle in radian measure into degrees. π/36 radians

Recall the relationship between the degrees and radians is 360°=2π radians and 180°=π radians. When converting​ units, the desired units need to be in the numerator and the units being cancelled need to be in the denominator.​ Therefore, to convert from radian measure to degree​ measure, multiply by 180°/π radians. Now convert the given angle from radians to degrees by multiplying by 180°/π radians. π/36 radians = π/36 radians• 180°/π radians Multiply and simplify. Note that the π radians cancels out. π/36 radians• 180°/π radians =180°/36 =5° ​ Therefore, π/36 radians= 5°.

What does a clockwise rotation of a ray mean in terms of its measure?

The measure is negative

What does a counterclockwise rotation of a ray mean in terms of its measure?

The measure is positive

Radian

The measure of a central angle that has an intercepted arc equal in length to the radius of the circle.

Converting Degrees to Radians Ex. Convert the following degree measure to radian measure. 85°

To make the​ conversion, use the fact that π radians=180°. This fact is used to form two conversion​ factors, π radians/180° and 180°/π radians. To convert from degrees to​ radians, the given angle should be multiplied by π radians/180° so that the degrees​ cancel, giving the answer in terms of radians. 85°• π radians/180° =85π/180 radians Now reduce the fraction to lowest terms. 85π/180 radians =5• 17π/5 •36 radians =17π/36 radians