• Tan θ = opposite leg/adjacent leg
• Tan θ = sin θ/cos θ = y/x
• This is because sin θ = opposite = y, cos θ = adjacent = x

• IaI = amplitude
• b = frequency (number of cycles in 360 degrees or 2π)
• h = horizontal shift
• k = vertical shift

• fequency * period = 360 degrees or 2π

sin(a+b) = sin(a) cos(b) + cos(a) sin(b)

cos(a+b) = cos(a) cos(b) - sin(a) sin(b)

sin(a-b) = sin(a) cos(b) - cos(a) sin(b)

cos(a-b) = cos(a) cos(b) + sin(a) sin(b)

sin(2a) = 2 sin(a) cos(a)

cos(2a) = cos^2(a) - sin^2(a) or

2 cos^2(a) -1 or

1 - 2 sin^2(a)

• The number of degrees/radians per unit of time
• The angular velocity of a point on a rotating object is the number of degrees through which the points turns per unit of time.
• When there are two rotating objects connected by an axle, the two objects will have the same angular velocity

• The distance per unit of time
• The linear velocity of a point on a rotating object is the distance the point travels along its circular path per unit of time.
• When there are two rotating objects connected by their rims, then the two objects will have the same linear velocity.
  • This can also be true for pulley systems, belts, chains, and gears.

ω = θ/t

where ω is angular velocity (which is often written in radians per unit of time), θ is the angle through which a point rotates, and t is the length of time it took to rotate through a particular angle

v = a/t

where v is the linear velocity (which is usually written in distance per unit of time), a is the number of units of arc length through which a point moves, and t is the length of time it took to rotate through a particular angle

a = rθ

v = rω

Use these equations when θ is in radians and ω is in radians per unit of time

The concepts that come along with the unit circle are endless. For the rest of your math careers, especially in this Pre-Calculus and Calculus class, you will need to master the skills of all trig functions to attack more complex problems in the future. In addition to the unit circle, the applications of sinusoids graphs and models are also endless and you can find them everywhere. For instance, if you look at sound wave, you will see a variation of a sinusoidal graph. Generally, sinusoidal graphs are used when regarding frequency, which encapsulates a lot of concepts nonetheless.

Like the unit circle and sinusoids, the concepts of rotary motion are also very applicable to the real-world, as well as other branches of mathematics and science, such as physics. Rotary motion allows students to understand the principles of rotating objects and the correlation between two spinning objects. Basically, whenever there are rotating objects involved and the objective is to find velocity or some aspect of the rotating object, rotary motion is involved, such as determining the speed of a bicycle's wheels. These problems can become much more complex as the math class advances. However, the concept of rotary motion can be connected to physics, as well, such as determining different velocities depending on a different radius and other factors.