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
For the unit circle, as theta, θ, increases from 0° to 90°, what happens to the length of the radius?

No change
The radius (the distance from the origin to the outside of the circle) does not change if theta changes.

Question 2
For the unit circle, as theta, θ, increases from 0° to 90°, what happens to the length of the rise (the line whose length is sin(θ))?

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
For the unit circle, as theta, θ, increases from 0° to 90°, what happens to the length of the run, (the line whose length is cos(θ))?

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
For the unit circle, between 0° and 90°:
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?

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:
AP Physics

AP Physics
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
How many degrees is π radians? How many radians make one complete circle?

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
What fraction of a circle is π/4 radians? π/6 radians? 33π/72 radians?

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
How many radians is one-quarter of a circle? Three-fourths of a circle? Two-thirds of a circle?

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:
AP Physics

Question 4
Convert each degree measure to radians: 90°, 270°, 30°, 45°, 60°.

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°):
AP Physics


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
Which of the following quantities are vectors and which are scalars?
Number of objects

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
Imagine that in your basement, you have a heavy safe. You push on the safe with all your might, and the safe does not move. Why not? You recruit four friends and all of you push on the safe, which starts to slide. Why does the safe now slide?

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.


Want more AP Physics practice? Check out more practice questions here!