FREE PREMIUM CONTENT<\/strong><\/h4>\n\n\n\nAP Physics Formulas & Equations Sheets<\/strong><\/h4>\n\n\n\nDownload a free list of formulas and equations you should know for the AP Physics 1, 2 & C exams. <\/p>\n<\/div>\n\n\n\n
\n\nDownload<\/a><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\nObjective 1<\/h3><\/div><\/div><\/div>
Describe the sine and cosine functions using the unit circle
Objective 1 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- The unit circle<\/em> is a circle with radius = 1, centered at point (0, 0)\u2014the origin <\/em><\/li>\n
- To use the unit circle, draw a radius from the origin to the circle at some angle theta,\u00a0\u03b8<\/em><\/li>\n
- The rise:\n
\n- Where the radius and the circle intersect, draw a line straight down to the x-axis<\/li>\n
- This line is called the rise<\/em> or the opposite <\/em><\/li>\n
- Its length is sin(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The run:\n
\n- Where the rise meets the x-axis, draw a line along the x-axis back to the origin<\/li>\n
- This line is called the run or the adjacent<\/li>\n
- Its length is cos(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The unit circle is a simple tool for understanding and remembering the relationships between common angles, sine values, and cosine values<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4><\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the radius?<\/p>\n<\/div><\/div>
Question 1 Answer <\/span><\/span><\/span><\/p> No change <\/strong>
The radius (the distance from the origin to the outside of the circle) does not change if theta changes.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the rise (the line whose length is sin(\u03b8<\/em>))?<\/p>\n<\/div><\/div>
Question 2 Answer <\/span><\/span><\/span><\/p> Increases <\/strong>
As theta increases, the length of the vertical line connecting the point where the radius and the circle intersect to the x-axis (the rise), increases. This length is equal to sin(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the run, (the line whose length is cos(\u03b8<\/em>))?<\/p>\n<\/div><\/div>
Question 3 Answer <\/span><\/span><\/span><\/p> Decreases <\/strong>
As theta increases, the horizontal line connecting the point where the radius and the circle intersect to the y-axis (the run), decreases. This length is equal to cos(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
For the unit circle, between 0\u00b0 and 90\u00b0:
A. At what angle is the rise the longest? The shortest?
B. At what angle is the run the longest? The shortest?
C. At what angle are the rise and the run the same length?<\/p>\n<\/div><\/div>
Question 4 Answer <\/span><\/span><\/span><\/p> A. sin(\u03b8<\/em>) is longest at 90\u00b0 and shortest at 0\u00b0;
B. cos(\u03b8<\/em>) is longest at 0\u00b0 shortest at 90\u00b0; 45\u00b0
C. Referring to our reasoning for the previous two questions, the sine of the angle will increase as theta increases, and cosine decreases as\u00a0\u03b8<\/em> increases. The 45\u00b0 angle is halfway between 0\u00b0 and 90\u00b0; therefore, its rise and run are equal.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 5<\/strong>
Fill in the following table:
![\"AP](\"http:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2018\/03\/Screen-Shot-2018-03-30-at-12.18.42-PM.png\")
<\/p>\n<\/div><\/div>
Question 5 Answer <\/span><\/span><\/span><\/p> ![\"AP](\"http:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2018\/03\/Screen-Shot-2018-03-30-at-12.19.28-PM.png\")
This table highlights the shortcut for calculating sine and cosine of standard angles. Note that the sine and cosine values for 45\u00b0 are the same.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\n
Objective 2<\/h3><\/div><\/div><\/div>
Convert between degrees and radians
Objective 2 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- Angles may be measured in degrees<\/em> or radians<\/em>, where radians are the SI unit for angles<\/li>\n
- Radians are defined using the concept of pi:\n
\n- Pi, \u03c0, is the ratio of a circle\u2019s circumference to its diameter <\/em><\/li>\n
- You can imagine diameter as a line passing straight from the right edge of the circle to the left edge of the circle<\/li>\n
- So a \u201c\u03c0 angle\u201d is actually a straight line<\/li>\n<\/ul>\n<\/li>\n
- Radians and \u03c0 are very useful for reporting angles as fractions:\n
\n- 2\u03c0 radians make one complete circle<\/li>\n
- Multiply 2\u03c0 by any fraction, and you find that fraction of a circle<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4><\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
How many degrees is \u03c0 radians? How many radians make one complete circle?<\/p>\n<\/div><\/div>
Question 1 Answer <\/span><\/span><\/span><\/p> 180\u00b0; 2\u03c0<\/strong>
\u03c0 is defined as the ratio of a circle\u2019s circumference to its diameter. The angle of a circle along its diameter (the line that cuts a circle in half) is 180\u00b0.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
What fraction of a circle is \u03c0\/4 radians? \u03c0\/6 radians? 33\u03c0\/72 radians?<\/p>\n<\/div><\/div>
Question 2 Answer <\/span><\/span><\/span><\/p> 1\/8; 1\/12; 11\/48 <\/strong>
Since an entire circle is 2\u03c0 radians and \u03c0\/4 = 2\u03c0\/8 = (1\/8)(2\u03c0), that means \u03c0\/4 is 1\/8 of a full circle. For the angles: \u03c0\/6 = 2\u03c0\/12 = (1\/12)(2\u03c0) and 33\u03c0\/72 = 22\u03c0\/48 = (11\/48)(2\u03c0).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
How many radians is one-quarter of a circle? Three-fourths of a circle? Two-thirds of a circle?<\/p>\n<\/div><\/div>
Question 3 Answer <\/span><\/span><\/span><\/p> \u03c0\/2; 3\u03c0\/2; 4\u03c0\/3<\/strong>
The number of radians in a fraction of a circle can be found by multiplying any fraction by 2\u03c0. For example:
![\"AP](\"http:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2018\/04\/Screen-Shot-2018-04-05-at-8.04.52-AM.png\")
<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
Convert each degree measure to radians: 90\u00b0, 270\u00b0, 30\u00b0, 45\u00b0, 60\u00b0.<\/p>\n<\/div><\/div>
Question 4 Answer <\/span><\/span><\/span><\/p> \u03c0\/2; 3\u03c0\/2; \u03c0\/6; \u03c0\/4; \u03c0\/3 <\/strong>
To calculate the number of radians in an angle, first calculate the fraction of a circle that makes up the angle, then multiply by 2\u03c0 to convert to radians. For the 90\u00b0 example, 90\u00b0 is 1\/4 of a circle (90\u00b0\/360\u00b0):
![\"AP](\"http:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2018\/04\/Screen-Shot-2018-04-05-at-8.08.18-AM.png\")
<\/a><\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\nObjective 3<\/h3><\/div><\/div><\/div>
Differentiate between vectors and scalars
Objective 3 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- In physics, many numbers have both magnitude and net direction\n
\n- These numbers, with magnitude and net direction, are called vectors<\/li>\n
- A vector\u2019s direction is usually given as an angle relative to the x-axis, or relative to another vector<\/li>\n
- The magnitude of a vector is just a number<\/li>\n
- All vectors in physics can be given in SI units<\/li>\n<\/ul>\n<\/li>\n
- In physics, some numbers do not have a net direction; these are called scalars<\/li>\n
- A vector is usually represented as an arrow\n
\n- The length of the arrow represents the vector\u2019s magnitude<\/li>\n
- The direction of the arrow is the same as the vector\u2019s direction<\/li>\n<\/ul>\n<\/li>\n
- Often, a vector\u2019s direction is ignored, so many vector quantities have a scalar version<\/li>\n<\/ul>\n
Objective 3 Practice Questions<\/h4><\/div><\/div><\/div>
\t
Download a free list of formulas and equations you should know for the AP Physics 1, 2 & C exams. <\/p>\n<\/div>\n\n\n\n
Objective 1<\/h3><\/div><\/div><\/div>
Describe the sine and cosine functions using the unit circle
Objective 1 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- The unit circle<\/em> is a circle with radius = 1, centered at point (0, 0)\u2014the origin <\/em><\/li>\n
- To use the unit circle, draw a radius from the origin to the circle at some angle theta,\u00a0\u03b8<\/em><\/li>\n
- The rise:\n
\n- Where the radius and the circle intersect, draw a line straight down to the x-axis<\/li>\n
- This line is called the rise<\/em> or the opposite <\/em><\/li>\n
- Its length is sin(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The run:\n
\n- Where the rise meets the x-axis, draw a line along the x-axis back to the origin<\/li>\n
- This line is called the run or the adjacent<\/li>\n
- Its length is cos(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The unit circle is a simple tool for understanding and remembering the relationships between common angles, sine values, and cosine values<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4><\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the radius?<\/p>\n<\/div><\/div>
Question 1 Answer <\/span><\/span><\/span><\/p> No change <\/strong>
The radius (the distance from the origin to the outside of the circle) does not change if theta changes.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the rise (the line whose length is sin(\u03b8<\/em>))?<\/p>\n<\/div><\/div>
Question 2 Answer <\/span><\/span><\/span><\/p> Increases <\/strong>
As theta increases, the length of the vertical line connecting the point where the radius and the circle intersect to the x-axis (the rise), increases. This length is equal to sin(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the run, (the line whose length is cos(\u03b8<\/em>))?<\/p>\n<\/div><\/div>
Question 3 Answer <\/span><\/span><\/span><\/p> Decreases <\/strong>
As theta increases, the horizontal line connecting the point where the radius and the circle intersect to the y-axis (the run), decreases. This length is equal to cos(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
For the unit circle, between 0\u00b0 and 90\u00b0:
A. At what angle is the rise the longest? The shortest?
B. At what angle is the run the longest? The shortest?
C. At what angle are the rise and the run the same length?<\/p>\n<\/div><\/div>
Question 4 Answer <\/span><\/span><\/span><\/p> A. sin(\u03b8<\/em>) is longest at 90\u00b0 and shortest at 0\u00b0;
B. cos(\u03b8<\/em>) is longest at 0\u00b0 shortest at 90\u00b0; 45\u00b0
C. Referring to our reasoning for the previous two questions, the sine of the angle will increase as theta increases, and cosine decreases as\u00a0\u03b8<\/em> increases. The 45\u00b0 angle is halfway between 0\u00b0 and 90\u00b0; therefore, its rise and run are equal.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 5<\/strong>
Fill in the following table:
<\/p>\n<\/div><\/div>
Question 5 Answer <\/span><\/span><\/span><\/p>
This table highlights the shortcut for calculating sine and cosine of standard angles. Note that the sine and cosine values for 45\u00b0 are the same.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\n
Objective 2<\/h3><\/div><\/div><\/div>
Convert between degrees and radians
Objective 2 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- Angles may be measured in degrees<\/em> or radians<\/em>, where radians are the SI unit for angles<\/li>\n
- Radians are defined using the concept of pi:\n
\n- Pi, \u03c0, is the ratio of a circle\u2019s circumference to its diameter <\/em><\/li>\n
- You can imagine diameter as a line passing straight from the right edge of the circle to the left edge of the circle<\/li>\n
- So a \u201c\u03c0 angle\u201d is actually a straight line<\/li>\n<\/ul>\n<\/li>\n
- Radians and \u03c0 are very useful for reporting angles as fractions:\n
\n- 2\u03c0 radians make one complete circle<\/li>\n
- Multiply 2\u03c0 by any fraction, and you find that fraction of a circle<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4><\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
How many degrees is \u03c0 radians? How many radians make one complete circle?<\/p>\n<\/div><\/div>
Question 1 Answer <\/span><\/span><\/span><\/p> 180\u00b0; 2\u03c0<\/strong>
\u03c0 is defined as the ratio of a circle\u2019s circumference to its diameter. The angle of a circle along its diameter (the line that cuts a circle in half) is 180\u00b0.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
What fraction of a circle is \u03c0\/4 radians? \u03c0\/6 radians? 33\u03c0\/72 radians?<\/p>\n<\/div><\/div>
Question 2 Answer <\/span><\/span><\/span><\/p> 1\/8; 1\/12; 11\/48 <\/strong>
Since an entire circle is 2\u03c0 radians and \u03c0\/4 = 2\u03c0\/8 = (1\/8)(2\u03c0), that means \u03c0\/4 is 1\/8 of a full circle. For the angles: \u03c0\/6 = 2\u03c0\/12 = (1\/12)(2\u03c0) and 33\u03c0\/72 = 22\u03c0\/48 = (11\/48)(2\u03c0).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
How many radians is one-quarter of a circle? Three-fourths of a circle? Two-thirds of a circle?<\/p>\n<\/div><\/div>
Question 3 Answer <\/span><\/span><\/span><\/p> \u03c0\/2; 3\u03c0\/2; 4\u03c0\/3<\/strong>
The number of radians in a fraction of a circle can be found by multiplying any fraction by 2\u03c0. For example:
<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
Convert each degree measure to radians: 90\u00b0, 270\u00b0, 30\u00b0, 45\u00b0, 60\u00b0.<\/p>\n<\/div><\/div>
Question 4 Answer <\/span><\/span><\/span><\/p> \u03c0\/2; 3\u03c0\/2; \u03c0\/6; \u03c0\/4; \u03c0\/3 <\/strong>
To calculate the number of radians in an angle, first calculate the fraction of a circle that makes up the angle, then multiply by 2\u03c0 to convert to radians. For the 90\u00b0 example, 90\u00b0 is 1\/4 of a circle (90\u00b0\/360\u00b0):
<\/a><\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\nObjective 3<\/h3><\/div><\/div><\/div>
Differentiate between vectors and scalars
Objective 3 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- In physics, many numbers have both magnitude and net direction\n
\n- These numbers, with magnitude and net direction, are called vectors<\/li>\n
- A vector\u2019s direction is usually given as an angle relative to the x-axis, or relative to another vector<\/li>\n
- The magnitude of a vector is just a number<\/li>\n
- All vectors in physics can be given in SI units<\/li>\n<\/ul>\n<\/li>\n
- In physics, some numbers do not have a net direction; these are called scalars<\/li>\n
- A vector is usually represented as an arrow\n
\n- The length of the arrow represents the vector\u2019s magnitude<\/li>\n
- The direction of the arrow is the same as the vector\u2019s direction<\/li>\n<\/ul>\n<\/li>\n
- Often, a vector\u2019s direction is ignored, so many vector quantities have a scalar version<\/li>\n<\/ul>\n
Objective 3 Practice Questions<\/h4><\/div><\/div><\/div>
\t
Describe the sine and cosine functions using the unit circle
Objective 1 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- The unit circle<\/em> is a circle with radius = 1, centered at point (0, 0)\u2014the origin <\/em><\/li>\n
- To use the unit circle, draw a radius from the origin to the circle at some angle theta,\u00a0\u03b8<\/em><\/li>\n
- The rise:\n
\n- Where the radius and the circle intersect, draw a line straight down to the x-axis<\/li>\n
- This line is called the rise<\/em> or the opposite <\/em><\/li>\n
- Its length is sin(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The run:\n
\n- Where the rise meets the x-axis, draw a line along the x-axis back to the origin<\/li>\n
- This line is called the run or the adjacent<\/li>\n
- Its length is cos(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The unit circle is a simple tool for understanding and remembering the relationships between common angles, sine values, and cosine values<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4><\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the radius?<\/p>\n<\/div><\/div>
Question 1 Answer <\/span><\/span><\/span><\/p> No change <\/strong>
The radius (the distance from the origin to the outside of the circle) does not change if theta changes.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the rise (the line whose length is sin(\u03b8<\/em>))?<\/p>\n<\/div><\/div>
Question 2 Answer <\/span><\/span><\/span><\/p> Increases <\/strong>
As theta increases, the length of the vertical line connecting the point where the radius and the circle intersect to the x-axis (the rise), increases. This length is equal to sin(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the run, (the line whose length is cos(\u03b8<\/em>))?<\/p>\n<\/div><\/div>
Question 3 Answer <\/span><\/span><\/span><\/p> Decreases <\/strong>
As theta increases, the horizontal line connecting the point where the radius and the circle intersect to the y-axis (the run), decreases. This length is equal to cos(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
For the unit circle, between 0\u00b0 and 90\u00b0:
A. At what angle is the rise the longest? The shortest?
B. At what angle is the run the longest? The shortest?
C. At what angle are the rise and the run the same length?<\/p>\n<\/div><\/div>
Question 4 Answer <\/span><\/span><\/span><\/p> A. sin(\u03b8<\/em>) is longest at 90\u00b0 and shortest at 0\u00b0;
B. cos(\u03b8<\/em>) is longest at 0\u00b0 shortest at 90\u00b0; 45\u00b0
C. Referring to our reasoning for the previous two questions, the sine of the angle will increase as theta increases, and cosine decreases as\u00a0\u03b8<\/em> increases. The 45\u00b0 angle is halfway between 0\u00b0 and 90\u00b0; therefore, its rise and run are equal.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 5<\/strong>
Fill in the following table:
<\/p>\n<\/div><\/div>
Question 5 Answer <\/span><\/span><\/span><\/p>
This table highlights the shortcut for calculating sine and cosine of standard angles. Note that the sine and cosine values for 45\u00b0 are the same.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\n
Objective 2<\/h3><\/div><\/div><\/div>
Convert between degrees and radians
Objective 2 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- Angles may be measured in degrees<\/em> or radians<\/em>, where radians are the SI unit for angles<\/li>\n
- Radians are defined using the concept of pi:\n
\n- Pi, \u03c0, is the ratio of a circle\u2019s circumference to its diameter <\/em><\/li>\n
- You can imagine diameter as a line passing straight from the right edge of the circle to the left edge of the circle<\/li>\n
- So a \u201c\u03c0 angle\u201d is actually a straight line<\/li>\n<\/ul>\n<\/li>\n
- Radians and \u03c0 are very useful for reporting angles as fractions:\n
\n- 2\u03c0 radians make one complete circle<\/li>\n
- Multiply 2\u03c0 by any fraction, and you find that fraction of a circle<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4><\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
How many degrees is \u03c0 radians? How many radians make one complete circle?<\/p>\n<\/div><\/div>
Question 1 Answer <\/span><\/span><\/span><\/p> 180\u00b0; 2\u03c0<\/strong>
\u03c0 is defined as the ratio of a circle\u2019s circumference to its diameter. The angle of a circle along its diameter (the line that cuts a circle in half) is 180\u00b0.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
What fraction of a circle is \u03c0\/4 radians? \u03c0\/6 radians? 33\u03c0\/72 radians?<\/p>\n<\/div><\/div>
Question 2 Answer <\/span><\/span><\/span><\/p> 1\/8; 1\/12; 11\/48 <\/strong>
Since an entire circle is 2\u03c0 radians and \u03c0\/4 = 2\u03c0\/8 = (1\/8)(2\u03c0), that means \u03c0\/4 is 1\/8 of a full circle. For the angles: \u03c0\/6 = 2\u03c0\/12 = (1\/12)(2\u03c0) and 33\u03c0\/72 = 22\u03c0\/48 = (11\/48)(2\u03c0).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
How many radians is one-quarter of a circle? Three-fourths of a circle? Two-thirds of a circle?<\/p>\n<\/div><\/div>
Question 3 Answer <\/span><\/span><\/span><\/p> \u03c0\/2; 3\u03c0\/2; 4\u03c0\/3<\/strong>
The number of radians in a fraction of a circle can be found by multiplying any fraction by 2\u03c0. For example:
<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
Convert each degree measure to radians: 90\u00b0, 270\u00b0, 30\u00b0, 45\u00b0, 60\u00b0.<\/p>\n<\/div><\/div>
Question 4 Answer <\/span><\/span><\/span><\/p> \u03c0\/2; 3\u03c0\/2; \u03c0\/6; \u03c0\/4; \u03c0\/3 <\/strong>
To calculate the number of radians in an angle, first calculate the fraction of a circle that makes up the angle, then multiply by 2\u03c0 to convert to radians. For the 90\u00b0 example, 90\u00b0 is 1\/4 of a circle (90\u00b0\/360\u00b0):
<\/a><\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\nObjective 3<\/h3><\/div><\/div><\/div>
Differentiate between vectors and scalars
Objective 3 Notes<\/h4><\/div><\/div><\/div><\/p>\n\n- In physics, many numbers have both magnitude and net direction\n
\n- These numbers, with magnitude and net direction, are called vectors<\/li>\n
- A vector\u2019s direction is usually given as an angle relative to the x-axis, or relative to another vector<\/li>\n
- The magnitude of a vector is just a number<\/li>\n
- All vectors in physics can be given in SI units<\/li>\n<\/ul>\n<\/li>\n
- In physics, some numbers do not have a net direction; these are called scalars<\/li>\n
- A vector is usually represented as an arrow\n
\n- The length of the arrow represents the vector\u2019s magnitude<\/li>\n
- The direction of the arrow is the same as the vector\u2019s direction<\/li>\n<\/ul>\n<\/li>\n
- Often, a vector\u2019s direction is ignored, so many vector quantities have a scalar version<\/li>\n<\/ul>\n
Objective 3 Practice Questions<\/h4><\/div><\/div><\/div>
\t
- \n
- The unit circle<\/em> is a circle with radius = 1, centered at point (0, 0)\u2014the origin <\/em><\/li>\n
- To use the unit circle, draw a radius from the origin to the circle at some angle theta,\u00a0\u03b8<\/em><\/li>\n
- The rise:\n
- \n
- Where the radius and the circle intersect, draw a line straight down to the x-axis<\/li>\n
- This line is called the rise<\/em> or the opposite <\/em><\/li>\n
- Its length is sin(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The run:\n
- \n
- Where the rise meets the x-axis, draw a line along the x-axis back to the origin<\/li>\n
- This line is called the run or the adjacent<\/li>\n
- Its length is cos(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- The unit circle is a simple tool for understanding and remembering the relationships between common angles, sine values, and cosine values<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4>
<\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the radius?<\/p>\n<\/div><\/div>Question 1 Answer <\/span><\/span><\/span><\/p>
No change <\/strong>
The radius (the distance from the origin to the outside of the circle) does not change if theta changes.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the rise (the line whose length is sin(\u03b8<\/em>))?<\/p>\n<\/div><\/div>Question 2 Answer <\/span><\/span><\/span><\/p>
Increases <\/strong>
As theta increases, the length of the vertical line connecting the point where the radius and the circle intersect to the x-axis (the rise), increases. This length is equal to sin(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
For the unit circle, as theta, \u03b8<\/em>, increases from 0\u00b0 to 90\u00b0, what happens to the length of the run, (the line whose length is cos(\u03b8<\/em>))?<\/p>\n<\/div><\/div>Question 3 Answer <\/span><\/span><\/span><\/p>
Decreases <\/strong>
As theta increases, the horizontal line connecting the point where the radius and the circle intersect to the y-axis (the run), decreases. This length is equal to cos(\u03b8<\/em>).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
For the unit circle, between 0\u00b0 and 90\u00b0:
A. At what angle is the rise the longest? The shortest?
B. At what angle is the run the longest? The shortest?
C. At what angle are the rise and the run the same length?<\/p>\n<\/div><\/div>Question 4 Answer <\/span><\/span><\/span><\/p>
A. sin(\u03b8<\/em>) is longest at 90\u00b0 and shortest at 0\u00b0;
B. cos(\u03b8<\/em>) is longest at 0\u00b0 shortest at 90\u00b0; 45\u00b0
C. Referring to our reasoning for the previous two questions, the sine of the angle will increase as theta increases, and cosine decreases as\u00a0\u03b8<\/em> increases. The 45\u00b0 angle is halfway between 0\u00b0 and 90\u00b0; therefore, its rise and run are equal.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 5<\/strong>
Fill in the following table:<\/p>\n<\/div><\/div>Question 5 Answer <\/span><\/span><\/span><\/p>
This table highlights the shortcut for calculating sine and cosine of standard angles. Note that the sine and cosine values for 45\u00b0 are the same.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\nObjective 2<\/h3>
<\/div><\/div><\/div>
Convert between degrees and radiansObjective 2 Notes<\/h4>
<\/div><\/div><\/div><\/p>\n- \n
- Angles may be measured in degrees<\/em> or radians<\/em>, where radians are the SI unit for angles<\/li>\n
- Radians are defined using the concept of pi:\n
- \n
- Pi, \u03c0, is the ratio of a circle\u2019s circumference to its diameter <\/em><\/li>\n
- You can imagine diameter as a line passing straight from the right edge of the circle to the left edge of the circle<\/li>\n
- So a \u201c\u03c0 angle\u201d is actually a straight line<\/li>\n<\/ul>\n<\/li>\n
- Radians and \u03c0 are very useful for reporting angles as fractions:\n
- \n
- 2\u03c0 radians make one complete circle<\/li>\n
- Multiply 2\u03c0 by any fraction, and you find that fraction of a circle<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
Objective 1 Practice Questions<\/h4>
<\/div><\/div><\/div>
\t\t\tQuestion 1<\/strong>
How many degrees is \u03c0 radians? How many radians make one complete circle?<\/p>\n<\/div><\/div>Question 1 Answer <\/span><\/span><\/span><\/p>
180\u00b0; 2\u03c0<\/strong>
\u03c0 is defined as the ratio of a circle\u2019s circumference to its diameter. The angle of a circle along its diameter (the line that cuts a circle in half) is 180\u00b0.<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 2<\/strong>
What fraction of a circle is \u03c0\/4 radians? \u03c0\/6 radians? 33\u03c0\/72 radians?<\/p>\n<\/div><\/div>Question 2 Answer <\/span><\/span><\/span><\/p>
1\/8; 1\/12; 11\/48 <\/strong>
Since an entire circle is 2\u03c0 radians and \u03c0\/4 = 2\u03c0\/8 = (1\/8)(2\u03c0), that means \u03c0\/4 is 1\/8 of a full circle. For the angles: \u03c0\/6 = 2\u03c0\/12 = (1\/12)(2\u03c0) and 33\u03c0\/72 = 22\u03c0\/48 = (11\/48)(2\u03c0).<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 3<\/strong>
How many radians is one-quarter of a circle? Three-fourths of a circle? Two-thirds of a circle?<\/p>\n<\/div><\/div>Question 3 Answer <\/span><\/span><\/span><\/p>
\u03c0\/2; 3\u03c0\/2; 4\u03c0\/3<\/strong>
The number of radians in a fraction of a circle can be found by multiplying any fraction by 2\u03c0. For example:<\/p>\n <\/div> <\/div> <\/div><\/section><\/div>
\t\t\tQuestion 4<\/strong>
Convert each degree measure to radians: 90\u00b0, 270\u00b0, 30\u00b0, 45\u00b0, 60\u00b0.<\/p>\n<\/div><\/div>Question 4 Answer <\/span><\/span><\/span><\/p>
\u03c0\/2; 3\u03c0\/2; \u03c0\/6; \u03c0\/4; \u03c0\/3 <\/strong>
To calculate the number of radians in an angle, first calculate the fraction of a circle that makes up the angle, then multiply by 2\u03c0 to convert to radians. For the 90\u00b0 example, 90\u00b0 is 1\/4 of a circle (90\u00b0\/360\u00b0):<\/a><\/p>\n <\/div> <\/div> <\/div><\/section><\/div><\/p>\nObjective 3<\/h3>
<\/div><\/div><\/div>
Differentiate between vectors and scalarsObjective 3 Notes<\/h4>
<\/div><\/div><\/div><\/p>\n- \n
- In physics, many numbers have both magnitude and net direction\n
- \n
- These numbers, with magnitude and net direction, are called vectors<\/li>\n
- A vector\u2019s direction is usually given as an angle relative to the x-axis, or relative to another vector<\/li>\n
- The magnitude of a vector is just a number<\/li>\n
- All vectors in physics can be given in SI units<\/li>\n<\/ul>\n<\/li>\n
- In physics, some numbers do not have a net direction; these are called scalars<\/li>\n
- A vector is usually represented as an arrow\n
- \n
- The length of the arrow represents the vector\u2019s magnitude<\/li>\n
- The direction of the arrow is the same as the vector\u2019s direction<\/li>\n<\/ul>\n<\/li>\n
- Often, a vector\u2019s direction is ignored, so many vector quantities have a scalar version<\/li>\n<\/ul>\n
Objective 3 Practice Questions<\/h4>
<\/div><\/div><\/div>
\t
- In physics, many numbers have both magnitude and net direction\n
- Radians are defined using the concept of pi:\n
- The unit circle is a simple tool for understanding and remembering the relationships between common angles, sine values, and cosine values<\/li>\n<\/ul>\n
- Its length is sin(\u03b8<\/em>)<\/li>\n<\/ul>\n<\/li>\n
- To use the unit circle, draw a radius from the origin to the circle at some angle theta,\u00a0\u03b8<\/em><\/li>\n