Radian & Degree Measure MATH 109 - Precalculus S. Rook

Overview Section 4.1 in the textbook: Angles Degree measure Radian measure Converting between degrees & radians

Angles

Angles Angle: describes the “space” between two rays that are joined at a common endpoint Recall from Geometry that a ray has one terminating side and one non-terminating side Can also think about an angle as a rotation about the common endpoint Start at OA (Initial side) End at OB (Terminal side)

Angles (Continued) If the initial side is rotated counter-clockwise θ is a positive angle clockwise θ is a negative angle

Angles in Standard Position An angle θ is in standard position if its: Initial side extends along the positive x-axis in reference to the Cartesian Plane Vertex is (0, 0) The “element of” symbol can be used to denote an angle in standard position e.g. means θ is in standard position with its terminal side in Quadrant III

Degree Measure

Angle Measure Angle Measure: expresses the size of an angle i.e. the space in between the initial and terminal sides in the direction of rotation Two common types of angle measures: Degrees Radians

Degree Measure 1 degree corresponds to (1⁄360) of a complete revolution starting from the initial side of an angle to its terminal side i.e. Can be viewed in terms of a circle Common degree measurements to be familiar with: 360° makes one complete revolution The initial and terminal sides of the angle are the same 180° makes one half of a complete revolution 90° makes one quarter of a complete revolution

Degree Measure (Continued) Angles that measure: Between 0° and 90° are known as acute angles Exactly 90° are known as right angles Denoted by a small square between the initial and terminal sides Between 90° and 180° are known as obtuse angles Complementary angles: two angles whose measures sum to 90° Supplementary angles: two angles whose measures sum to 180°

Degrees & Minutes Degrees can be broken down even further using minutes 1° = 60’ To convert from decimal degrees to degrees and minutes: Use the decimal portion of the angle Multiply by the appropriate conversion ratio Align the units in the ratio so the degrees will divide out, leaving the minutes To convert from degrees and minutes to decimal degrees: Use the minutes from the angle measurement Align the units in the ratio so the minutes will divide out, leaving the degrees

Sketching Angles in Standard Position (Example) Ex 1: Sketch each angle in standard position: a) 293° b) -115°

Complementary & Supplementary Angles (Example) Ex 2: Find: i) the complement ii) the supplement θ = 65°

Converting from Degrees to Minutes & Vice Versa (Example) Ex 3: Convert a) to degrees and minutes and convert b) to decimal degrees – approximate if necessary: a) θ = 232.55° b) θ = 17° 22’

Radian Measure

Motivation for Introducing Radians In some calculations, we require the measure of an angle (θ) to be a real number – we need a unit other than degrees This unit is known as the radian Many calculations tend to become easier to perform when θ is in radians Further, some calculations can be performed or even simplified ONLY if θ is in radians However, degrees are still in use in many applications so a knowledge of both degrees and radians is ESSENTIAL

Radians For θ = 1 radian, s = r Radian Measure: A circle with central angle θ and radius r which cuts off an arc of length s has a central angle measure of where θ is in radians i.e. How many radii r comprise the arc length s For θ = 1 radian, s = r

Radian Measure (Example) Ex 4: Find the radian measure of the central angle of a circle of radius r that subtends an arc length of s A radius of 27 inches and an arc length of 6 inches

Converting Between Degrees and Radians

Relationship Between Degrees and Radians Given a circle with radius r, what arc length s is required to make one complete revolution? Recall that the circumference measures the distance or length around a circle What is the circumference of a circle with radius r? C = 2πr Thus, s = 2πr is the arc length of one revolution and is the number of radians in one revolution Therefore, θ = 360° = 2π consists of a complete revolution around a circle

Relationship Between Degrees and Radians (Continued) Equivalently: 180° = π radians You MUST memorize this conversion!!! Technically, when measured in radians, θ is unitless, but we sometimes append “radians” to it to differentiate radians from degrees Like radians, real numbers are unitless as well

Converting from Degrees to Radians & Vice Versa To convert from degrees to radians: Multiply by the conversion ratio so that degrees will divide out leaving radians If an exact answer is desired, leave π in the final answer If an approximate answer is desired, use a calculator to estimate π To convert from radians to degrees: Multiply by the conversion ratio so that radians will divide out leaving degrees

Common Angles Need to become familiar with the degree and radian conversion between the following commonly used angle measurements: Deg Rad 0° 30° π⁄6 45° π⁄4 60° π⁄3 90° π⁄2 180° π 270° 3π⁄2 360° 2π

Converting from Degrees to Radians & Vice Versa (Example) Ex 5: Convert a) & b) to degrees and convert c) & d) to radians – leave π in the answer when necessary: a) b) c) d)

Coterminal Angles Two angles are coterminal if: BOTH are standard angles Share the SAME terminal side How can we obtain an angle coterminal to an angle θ? The second angle must terminate where θ terminates Recall that one complete revolution around a circle is 360° in degrees or 2π in radians

Coterminal Angles (Example) Ex 6: Do the following: a) Given θ = -190° find in degrees: i) two coterminal angles and ii) all angles coterminal to θ b) Given θ = π⁄8 find in radians: i) two coterminal angles and ii) all angles coterminal to θ

Summary After studying these slides, you should be able to: Draw an angle in standard position Find both the complement and supplement of an angle Convert between degrees & minutes and decimal degrees and vice versa Calculate the radian measure of a circle with radius r and subtended by an arc length s Convert between radians & degrees and vice versa Additional Practice See the list of suggested problems for 4.1 Next lesson Trigonometric Functions: The Unit Circle (Section 4.2)