Angles and the Unit Circle Section 13.2 Angles and the Unit Circle

“Standard Position” of an Angle an angle is made up of two rays, called the initial and terminal sides angles which are measured in trigonometry usually start from what is called “standard position”

“Standard Position” of an Angle when an angle is in “standard position”: - the initial side is on the positive x-axis (does not move) - the terminal side “opens up” - the vertex of the angle is located at the origin http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceAngles.htm

Measuring an Angle in “Standard Position” if the terminal side rotates COUNTERCLOCKWISE, then the angle is POSITIVE if the terminal side rotates CLOCKWISE, then the angle is NEGATIVE full rotation: 360° (however, the rotation can go on forever in either direction)

Angles and Quadrants… if the angle goes counterclockwise, then the terminal side passes through the quadrants in the following order: 1, 2, 3, 4 if the angle goes clockwise, then the terminal side passes through in the reverse order (4, 3, 2, 1)

Example 1: Quadrants, Measuring an Angle in Standard Position  

“Coterminal” Angles “coterminal” angles are angles that have the same terminal side, but different measurements there is an infinite number of coterminal angles for a given angle (both positive and negative) to find the measure of a coterminal angle, either add or subtract 360 to or from the angle measurement (if less than 360, you can also use the opposite signed angle needed to “complete the rotation”)

“Coterminal” Angles http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceAngles.htm

Example 2: “Coterminal Angles” state an angle which is coterminal to the given angle: A) 48° B) 310° C) -170° D) 575° E) -810°

“Quadrantal” Angles a “quadrantal” angle is created when the terminal side of an angle lies on an axis line (either x or y) question for you…What angles are considered to be quadrantal in “standard position”? quadrantal angles can be either positive or negative

“Reference” Angles a “reference” angle for a given angle is the ACUTE angle formed by the terminal side of the given angle and the x-axis http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceAngles.htm

How to Find Reference Angles… depends upon the quadrant in which the given angle is located: If the angle is POSITIVE and in… Ask the following question… Quadrant 1 Reference Angle = Given Angle Quadrant 2 What angle must I add to the given angle to make it equal 180 degrees? Quadrant 3 What angle must I subtract from the given angle to make it equal 180 degrees? Quadrant 4 What angle must I add to the given angle to make it equal to 360 degrees (“finish the circle”)?

How to Find Reference Angles… If the angle is NEGATIVE and in… Ask the following question… Same ? as POSITIVE quadrant… Quadrant 4 Reference Angle = Given Angle (no negative sign) Quadrant 1 Quadrant 3 What angle do I have to add to the given angle in order to equal -180 degrees? Quadrant 2 What angle do I have to subtract from the given angle in order to equal -180 degrees? What angle do I have to add to the given angle in order to equal -360 degrees?

Example 3: “Reference Angles” state the reference angle related to the given angle: A) 52° B) 335° C) -148° D) 571° E) -760°

30-60-90 and 45-45-90 Triangles from Geometry, the sides of a 30-60-90 triangle have the following ratio: - side opposite the 30° angle: x - side opposite the 60° angle: x√3 - side opposite the 90° angle (hypotenuse): 2x

30-60-90 and 45-45-90 Triangles from Geometry, the sides of a 45-45-90 triangle have the following ratio: - side opposite the 45° angles: x - side opposite the 90° angle (hypotenuse): x√2

The “Unit Circle” a “unit circle” is centered around the origin and has a radius of 1 unit

Finding the Cosine and Sine of an Angle (θ) a unit circle can be used to help find the cosine and sine values for that angle: - cos θ: x-coordinate of point where the terminal side intersects the unit circle - sin θ: y-coordinate of point where the terminal side intersects the unit circle

Finding the Cosine and Sine of an Angle (θ) to find the coordinates of the intersecting point: - draw the angle - determine whether a 30-60-90 or 45-45-90 triangle exists (reference angle used in this determination) - use the ratios of either triangle in computing the lengths of the unknown sides (will be vertical and horizontal legs; hypotenuse is ALWAYS 1 (radius of unit circle)) - vertical length: y-value - horizontal length: x-value

Example 4: Finding the Cosine and Sine of an Angle Find cos θ and sin θ for the following angles: A) 45° B) 120° C) -210°

Special Angles and the Unit Circle

Homework for this section… Assignment #1: pgs. 708-710, #’s 1-20, 37-49, 62b (#’s 37-44, also find the reference angles for each) Assignment #2: pgs. 708-710, #’s 21-28, 36, 50, 52, 69-71