AP Physics: Vectors and Trigonometry
In your prep for the AP Physics exam, make sure you have a solid understanding of fundamental topics like vectors and trigonometry. For a brush-up, check out the notes and practice questions below!
Objective 1
Describe the sine and cosine functions using the unit circle
Objective 1 Notes
- The unit circle is a circle with radius = 1, centered at point (0, 0)—the origin
- To use the unit circle, draw a radius from the origin to the circle at some angle theta, θ
- The rise:
- Where the radius and the circle intersect, draw a line straight down to the x-axis
- This line is called the rise or the opposite
- Its length is sin(θ)
- The run:
- Where the rise meets the x-axis, draw a line along the x-axis back to the origin
- This line is called the run or the adjacent
- Its length is cos(θ)
- The unit circle is a simple tool for understanding and remembering the relationships between common angles, sine values, and cosine values
Objective 1 Practice Questions
Question 1 Answer
No change
The radius (the distance from the origin to the outside of the circle) does not change if theta changes.
Question 2 Answer
Increases
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(θ).
Question 3 Answer
Decreases
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(θ).
Question 4 Answer
A. sin(θ) is longest at 90° and shortest at 0°;
B. cos(θ) is longest at 0° shortest at 90°; 45°
C. Referring to our reasoning for the previous two questions, the sine of the angle will increase as theta increases, and cosine decreases as θ increases. The 45° angle is halfway between 0° and 90°; therefore, its rise and run are equal.
Question 5
Fill in the following table:
Question 5 Answer
This table highlights the shortcut for calculating sine and cosine of standard angles. Note that the sine and cosine values for 45° are the same.
Objective 2
Convert between degrees and radians
Objective 2 Notes
- Angles may be measured in degrees or radians, where radians are the SI unit for angles
- Radians are defined using the concept of pi:
- Pi, π, is the ratio of a circle’s circumference to its diameter
- You can imagine diameter as a line passing straight from the right edge of the circle to the left edge of the circle
- So a “π angle” is actually a straight line
- Radians and π are very useful for reporting angles as fractions:
- 2π radians make one complete circle
- Multiply 2π by any fraction, and you find that fraction of a circle
Objective 1 Practice Questions
Question 1 Answer
180°; 2π
π is defined as the ratio of a circle’s circumference to its diameter. The angle of a circle along its diameter (the line that cuts a circle in half) is 180°.
Question 2 Answer
1/8; 1/12; 11/48
Since an entire circle is 2π radians and π/4 = 2π/8 = (1/8)(2π), that means π/4 is 1/8 of a full circle. For the angles: π/6 = 2π/12 = (1/12)(2π) and 33π/72 = 22π/48 = (11/48)(2π).
Question 3 Answer
π/2; 3π/2; 4π/3
The number of radians in a fraction of a circle can be found by multiplying any fraction by 2π. For example:
Question 4 Answer
π/2; 3π/2; π/6; π/4; π/3
To calculate the number of radians in an angle, first calculate the fraction of a circle that makes up the angle, then multiply by 2π to convert to radians. For the 90° example, 90° is 1/4 of a circle (90°/360°):
Objective 3
Differentiate between vectors and scalars
Objective 3 Notes
- In physics, many numbers have both magnitude and net direction
- These numbers, with magnitude and net direction, are called vectors
- A vector’s direction is usually given as an angle relative to the x-axis, or relative to another vector
- The magnitude of a vector is just a number
- All vectors in physics can be given in SI units
- In physics, some numbers do not have a net direction; these are called scalars
- A vector is usually represented as an arrow
- The length of the arrow represents the vector’s magnitude
- The direction of the arrow is the same as the vector’s direction
- Often, a vector’s direction is ignored, so many vector quantities have a scalar version
Objective 3 Practice Questions
Question 1 Answer
Scalars: Temperature, Revenue, Number, Pressure; Vectors: Force, Velocity
A vector quantity is one that has both magnitude and direction, while scalar quantities have magnitude but not net direction. Pressure, for example, is related to force but has no net direction since it acts in all directions at once.
Question 2 Answer
More force is being applied with the group of friends, to overcome the forces keeping the safe in place.
The force of (static) friction needs to be overcome in order for the safe to move. More force was applied by the group of friends than with the single person, and this larger force was enough to overcome the force of friction.