Trigonometry Basics. Angle. vertex at the and initial side along the. Standard position. Positive and Negative Angles

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1 Name: Date: Pd: Trigonometry Basics Angle terminal side direction of rotation arrow CCW is positive CW is negative vertex initial side Standard position vertex at the and initial side along the Positive and Negative Angles angle in standard position angle in standard position Angles lie in the quadrant in which the terminal side rests. Examples: In what quadrant do the angles lie? Is the angle positive or negative? Quadrant Adv Alg/Precal Trig Basics Notes 1

2 name given to an angle whose terminal side coincides with one of the axes Quadrantal angle Quadrantal angle measures for positive angles. -90 Quadrantal angle measures for negative angles. 2 or more angles that have the same terminal side. Finding coterminal angles add/subtract 360 to the given angle Coterminal angles Example: Find a positive and a negative angle coterminal with the given angle. a. 30 b. 170 c. 200 d. 400 e. 750 Worksheet #1 Trig Basics Adv Alg/Precal Trig Basics Notes 2

3 Degree-Minute-Second (DMS) Many navigational systems use the degree-minute-second o (DMS) format. DDD MM SS Example: o Some calculations, however, use a decimal format for degree measures. Decimal format is also easier to enter into calculators. Converting from DMS to Decimal Minutes Seconds Decimal Degrees Example: Given: o Convert to Decimal: Converting from Decimal to DMS Degrees: the whole degrees to the left of the decimal point Minutes: multiply the decimal degrees by 60; the minutes is the whole number to the left of the decimal point Seconds: multiply the decimal minutes by 60 and use the product rounded to 1 decimal place Example: Express in DMS Degrees: 235 Minutes: * 60 = Seconds: 0.02 * 60 = 1.2 Answer: = Worksheet #2 Trig Basics Adv Alg/Precal Trig Basics Notes 3

4 The fraction of a circle s circumference that is intercepted by a central angle. Arc length is measured in linear units (inches, meters, cm, feet, etc.). Arc Length arc length Another way to measure angles central angle Whenever an angle is used by itself (not inside a trig function), the angle MUST be in radian units NEVER degrees. Definition: The central angle made by taking the radius of a circle and wrapping it along the edge of the circle. A Length of arc AC = 2.75 cm Radian B = 1 radian C radius = 2.75 cm So, if a central angle is 1.5 radians, then the intercepted arc is 1.5 times the radius of the circle if the length of an intercepted arc is 2.3 times the radius, then the central angle is 2.3 radians. π radians = 180 degrees to radians, multiply by 180 (usually leave in terms Converting between Radians and Degrees of ) from radians to degrees, multiply by decimal places) (usually round to Adv Alg/Precal Trig Basics Notes 4

5 Example: Convert 225 to radians 5 Answer: 225 * radians Convert 3 radians to degrees Answer: radians * Worksheet #3 Trig Basics 2 or more angles with the same terminal side. In the diagram angle and angle are coterminal angles. Very useful when evaluating trigonometric functions (later). Finding coterminal angles. Coterminal Angles If angle is measured in degrees, add or subtract 360 If angle is measured in radians, add or subtract 2 Examples: Find a positive and a negative angle coterminal with the given angle. a. 75 b Adv Alg/Precal Trig Basics Notes 5

6 the angle between the terminal side and the x-axis are always positive and less than 90 Reference Angles useful when evaluating trig functions and solving trig equations (later) found depending on quadrant. For positive angles, Quadrant I: reference angle = angle Quadrant II: reference angle = 180 angle Quadrant III: reference angle = angle 180 Quadrant IV: reference angle = 360 angle 2 angle angle angle Examples: Find the reference angle for the following angles. a. 175 b c. 300 d. 7 4 s r where s is the arc length, r is the radius, and is the central angle (IN RADIANS) Arc Length Theorem Example: A circle has a radius of 5 feet. Find the arc length intercepted by a central angle measuring 2 radians. s r Formula s (5 feet)(2 radians) Substitution s 10 feet Evaluation Example: A central angle of 4 radians intercepts an arc with length of 25 meters. What is the radius of the circle? Adv Alg/Precal Trig Basics Notes 6

7 Sector Area 1 2 A r where A is the area of the sector, r is the 2 radius, and is the central angle (IN RADIANS) Example: The minute hand of a clock is 4 inches long. How much area will the minute hand sweep through in 20 minutes? 1 2 A r Formula 2 radians 2 Find θ: 60 minutes 20 minutes A (4 in) Substitution radians A in Evaluation 3 A in 2 Worksheet #4 Trig Basics Adv Alg/Precal Trig Basics Notes 7

8 I feel a need a need for speed! change in distance Linear speed time change in central angle ( ) Angular speed time change in revolutions Rotational speed time Conversions between speed units Conversions between speed units 1 1 revolution = 1 circumference (distance travelled) (rotational) (linear) 2 1 circumference = 2 radians (linear) (angular) 3 2 radians = 1 revolution (angular) (rotational) Hint on performing unit conversions: Multiply by the units you want the units you have Example: Convert 2.1 miles to feet 5280 feet ( units you want) 2.1 miles * 11,088 feet 1 mile ( units you have) Example: Convert 35 miles to inches 5280 feet ( units you want) 12 inches ( units you want) 35 miles * * 2,217,600 inches 1 mile ( units you have) 1 foot ( units you have) Adv Alg/Precal Trig Basics Notes 8

9 Example: A bicycle with wheels that have a radius of 20 inches is travelling at 15 miles per hour. How many revolutions per second are the wheels turning? Analysis: You are given a linear speed. The given linear units are miles so you need to convert to inches (units of the radius). Time is given in hours, but the final answer requires it to be in seconds so you need to convert hours to seconds. Finally, once the linear units and time units are set, convert from linear units to rotational units using relationship 1 above. Solution: 15 miles 1 hour 1 min 5280 ft 12 in 1 revolution 1 hour 60 min 60 sec 1 mile 1 ft 2π(20 in) = 1.03 rev/sec Original units Convert hours to seconds Convert linear units to same units as radius Convert linear speed to rotational speed Example: The radius of a CD is 2.25 in. The CD player rotates the CD at 400 rpm (revolutions per minute). What is the linear speed (in feet per second) of a speck of dust on the outer edge of the CD? Worksheet #5 Trig Basics Adv Alg/Precal Trig Basics Notes 9