Section 2.1 Angles and Their Measure

Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle

Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian.

r 1 radian

Theorem Arc Length For a circle of radius r, a central angle of radians subtends an arc whose length s is

Relationship between Degrees and Radians -> 1 revolution = 2 π radians -> 180 o = π radians

Announcements Test Friday (Jan 30) in lab, ARM 213/219, material through section 2.2 Sample test posted...link from course Web site Bring picture ID…you will need to scan your ID upon entering the lab You may use a calculator up to TI 86. You can’t your the book or notes

Section 2.2 Right Angle Trigonometry

A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legs of the triangle. c b a

Initial side Terminal side

a b c

The six ratios of a right triangle are called trigonometric functions of acute angles and are defined as follows: Function nameAbbreviationValue

Pathagorean Theorem a b c a 2 + b 2 = c 2

Find the value of each of the six trigonometric functions of the angle Adjacent c = Hypotenuse = 13 b = Opposite = 12

c b a

Pythagorean Identities The equation sin 2 θ + cos 2 θ along with tan 2 θ + 1 = sec 2 θ and 1 + cot 2 θ = csc 2 θ are called the Pythagorean identities.

More Identities Reciprocal Identities Quotient Identities

Complementary Angles Theorem Cofunctions of complementary angles are equal. Two acute angles are complementary if the sum of their measures is a right angle…90 degrees.

α β The angles α and β are complementary in a right triangle, α + β = 90 degrees. Complementary Angles in Right Triangles

Cofunctions DegreesDegrees RadiansRadians

Using the Complementary Angle Theorem Find the exact value (no calculator) of the following expressions.

so