Chapter 4 Trigonometric Functions

Angles Trigonometry means measurement of triangles. In Trigonometry, an angle often represents a rotation about a point. Thus, the angle θ shown is the result of rotating its initial ray to its terminal ray.

Standard Position of an Angle An angle whose vertex is the origin and whose initial side coincides with the positive x-axis is an angles in standard position.

Positive and Negative Angles Positive angles are generated by counterclockwise rotation. Negative angles are generated by clockwise rotation.

Labeling angles Angles are labeled with Greek letters such as alpha (  ) and beta (  ), and theta (  ), as well as uppercase letters.

Quadrantal Angles If the terminal ray of an angle in standard position lies in the first quadrant, the angle is said to be a first-quadrantal angle. The second-, third- and fourth- quadrant angles are similarly defined.

What is a radian? A common unit for measuring smaller angles is the degree, of which there are 360° in one revolution.

Illustration of Arc Length When an arc of a circle has the same length as the radius of the circle, the measure of the central angle, is by definition one radian. One radian is approximately degrees.

Section 4.1, Figure 4.6, Illustration of Six Radian Lengths There are 360° or 2  radians in one revolution. 360° = 2  radians 180° =  radians 90° =  /2 radians 45° =  /4 radians

Common Radian Angles

Conversion Formulas 1.To convert degrees to radians, multiply degrees by 2.To convert radians to degrees, mutliply radians by

“Leave in terms of Pi” Convert 250° to radians and radians to degrees.

List the Special Angles

Coterminal Angles Two angles are coterminal if they have the same initial and terminal rays. “Different name For the same thing”

Finding Coterminal Angles You can find an angle coterminal to a given angle by adding or subtracting ???? For the positive angle find a Positive coterminal angle. Negative coterminal angle.

Finding Coterminal Angles For the negative angle, find positive and negative coterminal angles. Find two angles, one positive and one negative, that are coterminal with the angle.

Complementary and Supplementary Angles Two positive angles are complementary (complements of each other) if their sum is 90° or radians. Two positive angles are supplementary (supplements of each other) if their sum is 180° or radians.

Arc length Arc length is measured in radians by the formula: Where S is arc length r is the radius is the central angle

A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°. A circle has a radius of 9 feet. Find the length of the arc intercepted by a central angle of /4.

Linear and Angular Speed Imagine a particle moving at a constant speed along the edge of a circle. The distance the particle travels with respect to time is the particle’s linear speed. Linear speed = Linear speed measures how fast the particle moves.

Linear and Angular Speed Imagine a spinning circle. Consider a central angle of that circle. Angular speed measures how fast the angle changes per a unit of time. Angular speed = Angular speed measures how fast the angle changes.

See examples on page 254 for Linear and Angular speed. Goes back to this idea

Classwork… Page 255 Know vocab odd (skip 21, 29) odd