Chapter 4 Trigonometric Functions Pre-Calculus Chapter 4 Trigonometric Functions

4.1 Radian & Degree Measure Objectives: Describe angles. Use radian measure. Use degree measure and convert between degree and radian measure. Use angles to model and solve real-life problems.

What is Trigonometry? Started as the measurement of triangles. Applications: astronomy, navigation, surveying. Developed into functions. Applications: sound waves, light rays, planetary orbits, vibrating strings, pendulums, orbits of atomic particles.

What is an Angle? An angle is formed by rotating a ray about its endpoint. Initial Side – starting position of the ray. Terminal Side – position after rotation. Vertex – endpoint of the ray. Usually labeled with Greek letters.

Angles in Standard Position Initial Side – lies on the positive x-axis Vertex – located at the origin. Positive Angle – counter- clockwise rotation (towards positive y-axis) Negative Angle – clock-wise rotation (towards negative y-axis)

Measure of an Angle The measure of an angle is the amount of rotation from the initial side to the terminal side. Angles are measured in radians or degrees.

Radians One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. Arc Length s = rθ, where θ is measured in radians

Radians and Degrees Degrees Radians 360° 2π 180° π 90° π/2 45° π/4 30° π/6 60° π/3 270° 3π/2 Circum. of a circle = 2πr One revolution = 360° Therefore, 2π = 360°

Angles in the Coordinate Plane

Co-terminal Angles Have the same initial and terminal sides. To find co-terminal angles, add or subtract multiples of 2π (or 360°).

Example 1 Find a positive and negative co-terminal angle for each and then sketch the angles.

Some Sketches for Example 1

Complementary & Supplementary Complementary Angles Supplementary Angles Note: Must be positive angles.

Example 2 If possible, find the complementary and supplementary angles for each.

Angle Conversions Degrees to Radians Radians = Degrees · Radians to Degrees Degrees = Radians · Note: π radians = 180°

Example 3 Convert from degrees to radians. 135° 540° –270°

Example 4 Convert from radians to degrees.

Example 5 A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240°.

Linear Speed Consider a particle moving at a constant speed along a circular arc of radius r. Let s be the length of the arc traveled in time t. The linear speed of the particle is given by That is, how fast is the particle moving along the arc?

Angular Speed Consider the same particle moving at a constant speed along the same circular arc of radius r. If θ is the angle (in radians) corresponding to the arc length s, then the angular speed of the particle is given by That is, how fast does the central angle change as the particle moves along the arc?

Example 6 The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of this second hand.

Example 7 A lawn roller with a 10-inch radius makes 1.2 revolutions per second. Find the angular speed of the roller in radians per second. Find the speed of the tractor that is pulling the roller.

Homework 4.1 Worksheet 4.1