Unit Circle

And Trigonometric Functions

(x, y) = (cos Ɵ, sin Ɵ)

Trigonometry relies on triangle proportionality.

Given right triangles with congruent acute angles, the trig function is built from the proportionality constant.

Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of the acute angle.

Angle Measurea. Degreesb. Radians

Radian is a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1.

Subtended Angle: The angle made by a line, arc or object.

Example: The Subtended Angle of the tree (from the person's point of view) is 22°

Measuring in Radians.

Trig Ratios

Trig Ratios

Trig Ratios

Trig Ratios

y = sin(x)

using a unit circle

Compare Graphs of sin and cos

Naming conventionAngles: Capital LettersSide lengths: Small Letter of Opposite Angle

Using Trig

What is the height of the tree on the left?

Using Trig

At 57" from the base of a building you need to look up at 55° to see the top of a building. What is the height of the building?

Using Trig

Find Reference Angles

Quadrant III

Quadrant IIQuadrant IV

Find Reference Angle or Reference Triangle

Sin 135°= Sin 45° =

Sin 300°= Sin -60° =

Angles greater than 360°

Reference Angle = 30°

Reference Angle = 55°

Positive and Negative Angles

Inverse Trig FunctionsWhen you want to find the angle measure Ɵ:

arcsin(x) = sin-1(x) Read as: “the angle whose sine is x”arccos(x) = cos-1(x)arctan(x) = tan-1(x)

The range of the Inverse Functions is limited as follows.

Inverse Trig Function

Here we a have a right triangle where we know the lengths of the two legs, that is, the sides opposite and adjacent to the angle. So, we use the inverse tangent function. If you enter this into a calculator set to "degree" mode, you get

If you have the calculator set to radian mode, you get

The base of a ladder is placed 3 feet away from a 10-foot-high wall, so that the top of the ladder meets the top of the wall. What is the measure of the angle formed by the ladder and the ground?