Trigonometric Functions Angles and Radian Measures

Objectives Identifying the parts of an angle Measuring angles using degrees and radians Calculating radian measure Conversion between degrees and radians Drawing angles in standard position Finding coterminal angles Calculating the length of a circular arc

Vocabulary Acute angle Ray Angle Reflex angle Coterminal angles Right angle Initial side Quadrant Negative angle Terminal side Obtuse angle Vertex Positive angle Quadrantal angle Radian Radian Measure

Angles A ray is a part of a line that has only one endpoint and extends forever in one direction. An angle is formed by two rays that have a common endpoint. The common endpoint is called the vertex. One ray is called the initial side, and the other ray is called the terminal side. Terminal side Initial side

An arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. Several methods can be used to name an angle. B 1 A C By using the end points with the vertex in the middle L CAB L BAC By using the letter of the vertex or a number L A L 1

An angle is in standard position if • its vertex is at the origin of a rectangular coordinate system and • its initial side lies along the positive x-axis. There are two types of rotation: (1) a counterclockwise rotation which results in a positive angle and (2) a clockwise rotation which yields a negative angle. Terminal side Positive x-axis Initial side When an angle is in standard position, its terminal side can lie in a quadrant. If the terminal side lies on the x-axis or y-axis, it does not lie in any quadrant. In this case it is called a quadrantal angle.

Measuring Angles Using Degrees We use the symbol º for degrees to indicate the measure of an angle. Certain angles have certain names based on their measures. 0°<𝜃<90° 𝜃=180° 𝜃=90° 180°<𝜃<360° 90°<𝜃<180°

Radian Measure Consider an arc of length s on a circle of radius r. The measure of the central angle, θ, that intercepts the arc is 𝜃= 𝑠 𝑟 A central angle in a circle with a radius of 6 inches intercepts an arc of length 15 inches. What is the radian measure of the central angle? 𝜃= 𝑠 𝑟 = 15 𝑖𝑛𝑐ℎ𝑒𝑠 6 𝑖𝑛𝑐ℎ𝑒𝑠 =2.5 𝑟𝑎𝑑𝑖𝑎𝑛𝑠=2.5

Relationship Between Degrees and Radians We know that 360 degrees is the amount of rotation of a ray back onto itself. Therefore we can come up with the following: 𝜃= 𝑠 𝑟 = 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙 𝑒 ′ 𝑠 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑟 = 2𝜋𝑟 𝑟 =2𝜋 One complete rotation measures 360º. One complete rotation measures 2𝜋 radians 360º = 2𝜋 radians 180º = 𝜋 radians

Conversion Between Degrees and Radians Using the basic relationship 𝜋 radians = 180º, To convert degrees to radians, multiply degrees by 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 180° To convert radians to degrees, multiply radians by 180° 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 Angles that are fractions of a complete rotation are usually expressed in radian measure as fractional multiples of 𝜋 instead of decimal approximations. We write 𝜃= 𝜋 2 rather than 𝜃≈1.57.

30° 90° −135° Remember: 𝜋=180° 30°∗ 𝜋 180° = 𝜋 6 90°∗ 𝜋 180° = 𝜋 2 Convert each angle in degrees to radians: 30° 90° −135° Remember: 𝜋=180° 30°∗ 𝜋 180° = 𝜋 6 90°∗ 𝜋 180° = 𝜋 2 −135°∗ 𝜋 180 ° = − 3𝜋 4

5𝜋 3 radians 𝜋 3 radians 𝜋=180°=3.14 radians 𝜋 3 ∗ 180° 𝜋 = 60° Convert each angle in radians to degrees: 𝜋 3 radians 5𝜋 3 radians 1 radian 𝜋=180°=3.14 radians 𝜋 3 ∗ 180° 𝜋 = 60° 5𝜋 3 ∗ 180° 𝜋 = 300° 1 radian∗ 180° 3.14 radians 57.32°

Degree and Angle Measures of Selected Positive and Negative Angles Positive angles Negative angles Degree and Angle Measures of Selected Positive and Negative Angles

Drawing Angles in Standard Position Draw and label each angle in standard position: 𝜃= 𝜋 4 𝛼= 5𝜋 4 𝛽=− 3𝜋 4 𝛾=− 9𝜋 4 Which quadrant does each angle lie in? II I Quadrant I Quadrant III Quadrant III III IV Quadrant IV

Coterminal Angles Two angles with the same initial and terminal sides but possibly different rotations are called coterminal angles. To find an angle that is coterminal, use the following formulas: 𝜃°±360°𝑘, where k is an integer or 𝜃±2𝜋𝑘, where k is an integer K is the number of rotations.

Terminal side Initial side

Terminal side Initial side

Find a positive angle that is less than 360º that is coterminal with each of the following: 𝜃°±360°𝑘, where k is an integer 420°−360°= 60° −120°+360°= 240° 400°−360°= 40° −135+360= 225°

17𝜋 6 angle − 𝜋 12 angle 13𝜋 5 angle − 𝜋 15 angle Find a positive angle that is less than 2𝜋 that is coterminal with each of the following: 17𝜋 6 angle − 𝜋 12 angle 13𝜋 5 angle − 𝜋 15 angle 𝜃±2𝜋𝑘, where k is an integer 17𝜋 6 −2𝜋= 17𝜋 6 − 12𝜋 6 = 5𝜋 6 − 𝜋 12 +2𝜋= − 𝜋 12 + 24𝜋 12 = 23𝜋 12 13𝜋 5 −2𝜋= 13𝜋 5 − 10𝜋 5 = 3𝜋 5 29𝜋 15 − 𝜋 15 +2𝜋= − 𝜋 15 + 30𝜋 15 =

The Length of a Circular Arc The length of the arc intercepted by the central angle is 𝑠=𝑟𝜃 A circle has a radius of 10 inches. Find the length of the arc intercepted by a central angle of 120º. Note: the angle, 𝜃, must be converted to radians. Then plug into the equation. 𝑠=10∗ 2𝜋 3 = 20𝜋 3 = 20∗3.14 3 ≈20.93 inches 𝜃=120°∗ 𝜋 180 °= 2𝜋 3

Linear speed is defined as the distance traveled for a given time. s is the arc length r is the radius t is the time 𝜽 is the angle expressed in radians 𝒗= 𝒔 𝒕 = 𝒓𝜽 𝒕

Linear Speed The second hand of a clock is 8 centimeters long. Find the linear speed of the tip of this second hand.

𝜔= 𝜃 𝑡 Angular speed is defined as the angle covered for a given time. 𝜽 the angle measured in radians t time

Angular Speed The circular blade on a saw rotates at 2400 revolutions per minute. Find the angular speed in radians per second.