1.1 Angles and Degree Measure

Transcription

1 J. Jenkins - Math 060 Notes. Angles and Degree Measure An angle is often thought of as being formed b rotating one ra awa from a fied ra indicated b an arrow. The fied ra is the initial side and the rotated ra is the terminal side. An angle whose verte is the center of a circle is a central angle, and the arc of the circle through which the terminal side moves is the intercepted arc. An angle in standard position is located in a rectangular coordinate sstem with the verte at the origin and the initial side on the positive -ais. terminal side α initial side verte α Angle α Central angle Angle in standard position Degree Measure of Angles The measure, m, of an angle, indicates the amount of rotation to the terminal side from the initial side. It is found b using an circle centered at the verte. A arc that forms complete circle arc is 360. The degree measure of an angle is the number of degrees in the intercepted arc of a circle centered at the verte. The degree measure is positive if the rotation is counterclockwise and negative if the rotation is clockwise. - - Unit Circle

2 acute obtuse straight quadrantal The initial angle ma be rotated in a positive or negative direction to get the position of the terminal side. If the terminal side is rotated for more than one revolution, it can form a coterminal angle, which is an angle with the same terminal side. Coterminal Angles - Angles and are coterminal if and onl if there is an integer k such that m m k360. Eample : Find two positive angles and two negative angles that are coterminal with 0. Eample : Determine whether angles in standard position with the given measures are coterminal. a) 690 and 390 Eample 3: Name the quadrant in which each angle lies. a) 55 b) 650 c) 360 Minutes and Seconds Historicall, angles were measured b using the degrees-minutes-seconds format, but with calculators it is convenient to have some fractional parts of degrees written in decimal form. Each degree is divided into 60 equal parts called minutes n, and each minute is divided into 60 equal parts called seconds n. A minute is 60 of a degree and a second is 60 of a minute or of a degree Eample 4: Convert the measure 35 5 to decimal degrees. Eample 5: Convert to degrees-minutes-seconds.

3 Eample 6: Perform the indicated operations. a) b) c) Radian Measure, Arc Length, and Area - - Unit Circle The radian measure of the angle in standard position is directed length of the intercepted arc on the unit circle. For a circle with radius r, the radian measure is the length of the intercepted arc divided b r. Because the circumference of the unit circle is C r, then the circumference of the unit circle is C (r ). Conversion from degrees to radians or radians to degrees is based on the formula 80 degrees radians Eample : Convert the degree measures to radians. a) 0 b) 7. Eample : Convert the radian measures to degrees. a) 5 b) 6. 7 radians 3 3

4 - - Unit Circle Eample 3: Find two positive and two negative angles using radian measure that are coterminal with 6. Arc Length of a Circle If a 30 central angle intercepts an arc length s on a circle of radius r, then a 60 central angle intercepts an arc of length s and a 90 angle intercepts an arc length of 3s. Since a central angle of 360 intercepts an arc whose length is the circumference of the circle, we have arc length circumference m in radians radians m in degrees 360 degrees The length s of an arc intercepted b a central angle of radians on a circle of radius r is given b s r Eample 4: a) A central angle of intercepts an arc on the surface of the earth that runs from the equator to the North Pole. Using 3950 miles as the radius of the earth, find the length of the intercept arc to the nearest mile. b) The wagon wheel has a diameter of 8 inches and an angle of 30 between the spokes. What is the length of the arc s (to the nearest hundredth of an inch) between two adjacent spokes? 4

5 Eample 5: The distance between Los Angeles and New York Cit is 4505 miles. Find the central angle to the nearest tenth of a degree that intercepts an arc of 4505 miles on the surface of the earth (radius 3950 miles). Area of a Sector of a Circle area of sector m in radians area of circle radians m in degrees 360 degrees Area of a Sector: The area A of a sector with central angle (in radians) in a circle of radius r is given b A r Eample 6: A center-pivot irrigation sstem is used to water a circular field with radius 00 feet. In three hours the sstem waters a sector with a central angle of. What area (in 8 square feet) is watered at that time?.3 Angular and Linear Velocit If the speedometer on our car sas 50 mph, then our velocit is 50 mph. Velocit is the rate at which the location of an object is changing with respect to time. We will discuss two tpes of velocit. The angular velocit of a point is the rate at which the angle is changing. The linear velocit of a point in motion is the rate at which the distance is changing. Angular Velocit 400 revolutions per minute could be considered angular velocit, however we will epress angular velocit in radians per unit of time. In order to convert from revolutions to radians, we use the fact that rad rev. 400 rev 400 rev rad 800 rad 53 rad/ min min min rev min We use the greek letter (omega) to represent angular velocit. Definition: If a point is in motion on a circle through an angle of radians in time t, then its angular velocit is given b t Eample : Changing the units Convert the angular velocit of 3000 rad/hr to rad/ sec. 5

6 Eample : Finding angular velocit A tpical lawnmower blade rotates at a rate of 350 revolutions per minute (rpm). What is the angular velocit in radians per second of a point on the tip of the blade? Linear Velocit If a point is in motion on a circle then the distance traveled b the point in a unit of time is the length of an arc on the circle. We use the letter v to represent linear velocit for a point in circular motion. Definition: If a point is in motion on a circle of radius r through an angle of radians in time t, then its linear velocit is given b v s t r t Eample 3: Linear Velocit of a propeller A propeller with a radius of 8 meters is rotating at 6 revolutions per second. What is the linear velocit in meters per second for a point on the tip of the propeller? Eample 4: What is the linear velocit in miles per hour of the tip of a 4 inch lawnmower blade that is rotating at 000 revolutions per minute? Linear Velocit in Terms of Angular Velocit Linear velocit is arc length over time v s/t and angular velocit is angle over time. Since s r, we have t v s t r r t r t We have found the relationship between the two velocities. Theorem: If v is the linear velocit of a point on a circle of radius r, and is its angular velocit, then v r An point on the surface of the earth (eept at the poles) makes one revolution radians about the ais of the earth is 4 hours. So the angular velocit of a point on the earth is 4 radians per hour. The linear velocit of a point on the surface of the earth depends on or its distance for the ais of the earth. Eample 5: Linear velocit on the surface of the earth What is the linear velocit in feet per second for a point on the equator? 6

7 .4 The Trigonometric Functions The si trigonometric functions are the sine, cosine, tangent, cosecant, secant, and cotangent functions. Abbreviations for the functions are sin, cos, tan, csc, sec, and cot, respectivel. There are several was to define these functions of trig. Definition of the Trigonometric Functions Suppose that is an angle in standard position whose terminal side contains the point,. Let r be the distance between, and the origin. Using the distance formula, r. The trigonmetric ratios are the si possible ratios that can be formed with the numbers,, and r, provided that no denominator is zero. (,) *Remember SOH CAH TOA r α The Trigonometric Fuctions sin opp hp r csc hp opp r cos adj hp r sec hp tan opp adj cot opp adj adj r Note that csc, cos, and tan are reciprocals for sin, cos, and tan, respectivel. Reciprocal Identities csc sec sin cos cot tan Eample : Evaluating the trig functions Find the values of the si trigonometric functions of the angle in standard position whose terminal side passes through the point,4. To evaluate trig functions for an angle, we must have a point, on the terminal side of the angle. The trig functions have the same values regardless of which point is used. If, and, are two points on the terminal side, then the two triangles formed are similar. Because similar triangles are proportional, we get the same values for the trig functions. 7

8 (',') r' (,) r ' α ' e) sin r r Eample : Evaluating trig functions for quadrantal angles Find the eact values. a) csc90 b) tan80 c) sin 3 d) cot0 Quadrants Find the signs of each function in the quadrants A good mnemonic for the signs of the basic function is "All Students Take Calculus" Trigonometric Functions at Multiples of triangle sin 45 cos45 tan 45 csc 45 sec 45 cot 45 8

9 Eample 3: Find the eact value of each function. a) sin 3 b) tan c) cos Trigonometric Functions at Multiples of 30 A triangle is made b making two congruent triangles formed b the altitude of an equilateral triangle

10 Eample 4: Find the eact value of each function a) sin60 b) cos 7 6 c) tan Eample 5: Find each function value rounded to four decimal places. a) sin3. 5 b) cos85. 3 c) tan 9 d) csc4. 5 e) sec3. 5 f) cot49. 0

11 .5 Right Triangle Trigonometr Inverse Trigonometric Functions A solution to the equation sin is an angle whose sine is. Because sin 30 and sin 50, could be 30 or 50. Since an angle with the same terminal side as 30 or 50 is a solution, there are infinitel man solutions. (Since right triangles other two angles are onl acute angles, we are onl interested in acute angle measures). Eample : Find the angle that satisfies each equation where a) cos b) sin c) tan In order to solve for algebraicall, take the inverse of both sides of the equation. Taking the inverse works for sin, cos, and tan equations. 3 e) sin sin sin sin 3 60 Definition of Inverse Trigonometric Functions sin provided sin and cos provided cos and 0 80 tan provided tan and Eample : Evaluate each epression. Give the result in degrees. a) sin 3 b) cos c) tan 3 Eample 3: Evaluate each epression. Give the result in degrees to the nearest tenth. a) sin 3/7 b) cos c) tan 3. 4 Right Triangles The trigonometric fuctions of an acute angle of a right triangle sin opp hp r csc hp opp r r cos adj hp r sec hp adj r tan opp adj cot opp adj

12 Eample 4: Find all si trigonometric functions for the angle. α c 5 Eample 5: Solve the right triangle (find all missing sides and angles) β 60 b a Eample 6: Solve the right triangle given a and b 5. Find the acute angles to the nearest tenth of a degree. Applications Using trigonometr, we can find the size of an object without actuall measuing the object. Two common terms used in regard are angle of elevation and angle of depression. β angle of depression α angle of elevation horizontal line

13 Eample 7: The angle of elevation of the top of a cell phone tower is 38. at a distance of 344 feet from the tower. What ist he height of the tower? Eample 8: At one location, the angle of elevation of the top of an antenna is 44.. At a point that is 00 feet closer to the antenna, the angle of elevation is 63.. What is the height of the antenna? Eample 9: A sailor spots a small island on the horizon from the top of a 40 ft mast. How far is it to the island? Use 3950 miles for the radius of the earth. Eample 0: Graphing Calculator Required It is about 00 miles from Houston to San Antonio on interstate I-0. A pilot at an altitude of 5 miles over Houston spots San Antonio on the horizon. From this information calculate the radius of the earth. 3

14 .6 The Fundamental Identit and Reference Angles The fundamental identi involves the squares of sine and cosine functions. For convenience, we write sin as sin and cos as cos. B definition sin r, cos r, and r. So sin cos r 3 r r r r Fundamental Identit of Trigonometr: sin cos Eample : Find sin if cos 4 and is an angle in quadrant IV. Reference Angle: If is a nonquadrantal angle in standard position, then the reference angle for is the positive acute angle (reads "theta prime") formed b the terminal side of and the positive or negative -asis. θ'= π - θ θ θ' θ' θ θ θ'= θ - π θ θ'= π - θ θ' θ' 4

15 Eample : For each angle, sketch the reference angle and give the measure of in both radians and degrees. a) 50 b) 5 c) 4 7 d) Eample 3: For each angle, find the sine and cosine using the reference angle. a) 50 b) 5 c) 4 7 d) Modeling with the Sine Function The trigonometric functions can be used to model periodic phenomena. Eample 4: The tide in the Ba of Fund rises and falls ever 3 hours. The depth of the water at a certain point in the ba is modeled b the function d 8 sin t 9, where t is 3 time in hours and d is depth in meters. Find the depth at t 3 (high tide) and t (low tide)

16 Chapter Review. Find a coterminal angle in radians or degrees m m k360 or m m k. Convert from decimal degrees to degrees, minutes, seconds Convert from degrees, minutes, seconds to decimal degrees minute 60 degree second 60 minute 3600 degree 3. Add, subtract, multipl, and divide degrees, minutes, seconds. 4. Convert from radians to degrees and degrees to radians. 80 degrees radians 5. Know and understand how to use arc length circumference 6. Find the arc length s r given that is in radians. m in radians radians m in degrees 360 degrees 7. Find the area of a sector A r 8. Convert from revolutions to radians or degrees rad 360 rev 9. Use conversions to change units (meters, inches, feet, miles, seconds, minutes, hours, etc.) 0. Find angular velocit t. Find linear velocit v s t r r t. The Trigonometric Fuctions sin opp hp r cos adj hp r csc opp hp r sec hp adj r r tan opp adj cot opp adj 3. Reciprocal Identities csc sec sin cos cot tan 4. Evaluate eact values of sin, cos, tan, csc, sec, and cot using reference angles. 5. Evaluate decimal values of sin, cos, tan, csc, sec, and cot using a calculator. 6. Know and understand the special right triangles and Know and understand the Unit Circle ( degrees, radians, ordered pairs) 8. Find an angle using trig functions, the unit circle, inverse functions, and a calculator. 9. Know and understand the signs of the trig functions in each quadrant (ASTC). 0. Solve a right triangle (find all missing sides and angles). Solve problems using angle of elevation and angle of depression.. Know and understand the fundamental identit of trigonometr sin cos 3. Find functions using a reference angle and the correct quadrant signs. 4. Know how to solve problems given formulas of a periodic phenomena. (sine wave) 6