Intro. To Unit Circle

The ray on the x-axis is the initial side of the angle

the other ray is the terminal side of the angle.

There are six different trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent.

Each of these functions represents the ratio between two sides of any right triangle. These ratios do not depend on the size of the right triangle, but only on the measures of the acute angles inside the triangle.

Rotation from the initial side all the way around to the initial side again is 360°.

One degree represents 1 over 360 of a rotation around the vertex.

Degrees to Radians radians = degrees • pi over 180 degrees

Radians to Degrees degrees = radians • 180 degrees over pi

The measure of an angle is positive when the rotation from the initial side to the terminal side is in the counterclockwise direction.

The measure is negative when the rotation is clockwise.

TOA tan θ = opposite leg over adjacent leg equals y over x

The saying SOH-CAH-TOA can be used to remember the three basic trigonometric ratios. If you use a triangle with a vertex as a central angle on the unit circle, the ratio can be expressed in terms of x and y. The symbol theta (θ) is used to represent the measure of an angle in standard position.

Because sin θ = y over r and the radius of the unit circle is 1, the sine of θ is the y-coordinate of the point at which the terminal side of the angle intersects the unit circle. Because cos θ = x over r and the radius of the unit circle is 1, the cosine of θ is the x-coordinate when θ is on the unit circle.

The tangent of θ is the ratio of the y-coordinate to the x-coordinate of the point on the unit circle corresponding to θ.

Since a radian is defined to be the ratio of the arc length and the radius, the equation below can be used to find arc length:

arc length: s=rθ

A radian is defined

as the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle.

sin θ = opposite leg over hypotenuse equals y over r

cos θ = adjacent leg over hypotenuse equals x over r

A central angle

is an angle with a vertex at the center of a circle and sides that are radii of the circle.

An angle is in standard position when

the vertex is at the origin and one ray is on the positive x-axis.