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Chapter 5

Chapter 5. Trigonometric Functions. Section 5.1. Angles and Degree Measure Learn how to convert decimal degree measures to degrees, minutes, and seconds. Find the number of degrees in a given number of rotations Identify angles that are coterminal with a given angle.

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Chapter 5

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  1. Chapter 5 Trigonometric Functions

  2. Section 5.1 • Angles and Degree Measure • Learn how to convert decimal degree measures to degrees, minutes, and seconds. • Find the number of degrees in a given number of rotations • Identify angles that are coterminal with a given angle

  3. 5.1 Angles and Degree Measure • Vertex • Endpoint of an angle • Initial Side • The ray of the angle that is fixed • Terminal Side • The second ray that rotates to form the angle • Standard Position • An angle with its vertex at the origin and its initial side along the positive x-axis

  4. 5.1 Angles and Degree Measure

  5. 5.1 Angles and Degree Measure • Degree • Most common unit to measure an angle • Minutes • A degree is subdivided into 60 equal parts called minutes (1’) • Seconds • A minute is subdivided into 60 equal parts called seconds (1”) • Example 1: • Change 15.735 degrees to degrees, minutes, and seconds. • 15.735o=15o+(0.735*60)’ • =15o+44.1’ • =15o+44’+(0.1*60)” • =15o+44’+6”

  6. 5.1 Angles and Degree Measure • Example 2: • Change 329.125 degrees to degrees, minutes, and seconds. • 329o+7’+30” • Example 3: • Change 39o+5’+34” to degrees • 39o+5’+34”=39o+5’*(1 o/60’)+34” (1 o/3600’) • 39.093o • Example 4: • Change 35o+12’+7” to degrees • 35.202o

  7. 5.1 Angles and Degree Measure • Quadrantal Angle • If the terminal side of an angle that is in standard position coincides with one of the axes • How many quadrantal angles are there? • 4 • What are their measures? • 90 degrees • 180 degrees • 270 degrees • 360 degrees

  8. 5.1 Angles and Degree Measure • Give the angle measure represented by the rotation about the axis • 5.5 Rotations clockwise • Which way do we rotate to go clockwise? • Negative, clockwise rotations ALWAYS have negative measures • 5.5 * -360 = -1980 degrees • 3.3 Rotations counterclockwise • Which way do we rotate to go counterclockwise? • Positive, counterclockwise rotations ALWAYS have positive measures • 3.3*360 = 1188 degrees • 9.5 Rotations clockwise • -3420 degrees • 6.75 Rotations counterclockwise • 2430 degrees

  9. 5.1 Angles and Degree Measure • Coterminal Angles • If α is the degree measure of an angle, then all angles measuring α+ 360k degrees, where k is an integer; are coterminal with α. • Two angles in standard position that have the same initial side • Identify all angles that are coterminal with a 45 degree angle. Find one positive and one negative angle that are also coterminal. • All angles having a measure of 45o+ 360ko • Positive Angle • 45o+ 360*(1)o=405o • Negative Angle • 45o+ 360*(-2)o=-675o

  10. 5.1 Angles and Degree Measure • Identify all angles that are coterminal with a 294 degree angle. Find one positive and one negative angle that are also coterminal. • All angles having a measure of 294o+ 360ko • Positive Angle • 294o+ 360*(1)o=654o • Negative Angle • 294o+ 360*(-1)o=-66o

  11. 5.1 Angles and Degree Measure • If an angle with 775 degrees is in standard position, determine a coterminal angle that is between 0 and 360 degrees. State the quadrant in which the terminal side lies. • Find the number of rotations about the axis by dividing 775 by 360 = 2.15278 • Since we need an angle between 0 and 360 degrees, what rotation number should we use? • Subtract 2 and use .15278 • α= .15278*360 • Could also take the number and continue to subtract 360 until you get a number between 0 and 360. • 55o • What quadrant does the terminal side fall in? • Quadrant 1

  12. 5.1 Angles and Degree Measure • If an angle with -777 degrees is in standard position, determine a coterminal angle that is between 0 and 360 degrees. State the quadrant in which the terminal side lies. • 303o • What quadrant does the terminal side fall in? • Quadrant 4

  13. 5.1 Angles and Degree Measure • Reference Angle • The acute angle formed by the terminal side of the given angle and the x-axis • Reference Angle Rule • For any angle α, 0o< α <360o, its reference angle α’ is defined by: • α, when the terminal side is in Quadrant I. • 180o- α, when the terminal side is in Quadrant II. • α – 180o, when the terminal side is in Quadrant III. • 360o- α, when the terminal side is in Quadrant IV.

  14. 5.1 Angles and Degree Measure • Find the measure of the reference angle for an angle with a measurement of 120 degrees • What Quadrant does this angle’s terminal side fall in? • Between 90o and 180o • Quadrant II • We use which formula? • 180o- α • 180o-120o=60o • Find the measure of the reference angle for an angle with a measurement of -135degrees • First we have to find a positive coterminal angle • 360o-135o = 225o • What quadrant does this angle’s terminal side fall in? • Between 180o and 270o • Quadrant III • α – 180o • 225o-180o=45o

  15. 5.1 Angles and Degree Measure • Find the measure of the reference angle for an angle with a measurement of 312 degrees • 48o • Find the measure of the reference angle for an angle with a measurement of -195 degrees • 15o

  16. Section 5.2 • Trigonometric Ratios in Right Triangles • Learn how to find the values of trigonometric ratios for actue angles of right triangles

  17. 5.2 Trigonometric Ratios in Right Triangles • What is a Right Triangle? • A triangle with a 90 degree angle in it. • How can we classify the other two angles in the right triangle? • Must be acute • Are also complementary • What are the parts of a Right Triangle? < 1 + < 2 = 90o 1 Side C is the hypotenuse Sides A and B are the legs 2

  18. 5.2 Trigonometric Ratios in Right Triangles • When looking at one specific acute angle in a triangle, we can classify the legs by: • Adjacent Side: the leg that is a side of the acute angle • Opposite Side: the leg that is the side opposite the angle • Looking at Triangle ABC, what are the adjacent and opposite sides for < B? B Hypotenuse Adjacent Opposite

  19. 5.2 Trigonometric Ratios in Right Triangles • Trigonometric Ratios • The ratios of the sides of right triangles based on a specific acute angle within the right triangle • Easy way to remember Sin, Cosine, and Tangent Ratios • SOHCAHTOA • Sine • Ratio of the side opposite Θ and the hypotenuse • Cosine • Ratio of the side adjacent Θ and the hypotenuse • Tangent • Ratio of the side opposite Θ and the side adjacent to Θ

  20. 5.2 Trigonometric Ratios in Right Triangles

  21. 5.2 Trigonometric Ratios in Right Triangles • Find the values of sine, cosine, and tangent for <B. • What do we need to do first? • Find the third side’s length • How do we do that? • Pythagorean Theorem, A2 + B2 = C2 • C=3*(85)1/2 • Which angle is our Θ? • Sin Θ • Opposite/hypotenuse • 18/33 • Cos Θ • Adjacent/hypotenuse • 3*(85)1/2/33 • Tan Θ • Opposite/Adjacent • 18/3*(85)1/2 18 m A C 3*(85)1/2 m 33 m Θ B

  22. 5.2 Trigonometric Ratios in Right Triangles • Find the values of sine, cosine, and tangent for <A. • Find the third side’s length • C=17 m • Which angle is our Θ? • Sin Θ • 8/17 • Cos Θ • 15/17 • Tan Θ • 8/15 C 17 m 8 m Θ A B 15 m

  23. 5.2 Trigonometric Ratios in Right Triangles • Cosecant • Opposite of sine, cscΘ = 1/sin Θ • Hypotenuse/side opposite • Secant • Opposite of cos, secΘ = 1/cosΘ • Hypotenuse/side adjacent • Cotangent • Opposite of tan, cotΘ = 1/tanΘ • Side Adjacent/ Side opposite

  24. 5.2 Trigonometric Ratios in Right Triangles • If cos Θ = ¾, what is sec Θ? • sec Θ = 1/cosΘ • =1/(3/4) • =4/3 • If sin Θ=0.8, what is csc Θ? • =1.25 • If csc Θ = 1.345 find sin Θ? • sin Θ = 1/cscΘ • =1/1.345 • =.7435 • If cot Θ = 6/5, what is tan Θ? • =5/6

  25. 5.2 Trigonometric Ratios in Right Triangles • Find the values of the six trigonometric ratios for <E. • First find third side using pythagorean theorem • (58)1/2 m • CosE = 3* (58)1/2 /58 • SinE= 7* (58)1/2 /58 • Tan E=7/3 • Sec E= (58)1/2 /3 • CscE= (58)1/2 /7 • Cot E=3/7 D (58)1/2 m 7 m Θ E F 3 m

  26. 5.2 Continued • Special Triangles • 30O-60O-90O • 45O-45O-90O • What are the special relationships we know about these triangles? • 30O-60O-90O

  27. 5.2 Continued • What are the special relationships we know about these triangles? • 45O-45O-90O

  28. 5.2 Continued Trigonometric Ratios for 30o, 60o, 90o

  29. 5.2 Continued • Cofunctions • Trigonometric functions that are equal when their arguments are complementary angles, such as sine and cosine, tangent and cotangent, and secant and cosecant. • Sin Θ = Cos (90O- Θ) • Cos Θ = Sin (90O- Θ) • Tan Θ=Cot (90O- Θ) • Cot Θ=Tan (90O- Θ) • Sec Θ=Csc (90O- Θ) • Csc Θ=Sec (90O- Θ)

  30. Section 5.3 Trigonometric Functions on the Unit Circle Find the values of the six trigonometric functions using the unit circle Find the values of the six trigonometric functions of an angle in standard position given a point on its terminal side

  31. Trigonometric Functions on the Unit Circle • Unit Circle • A circle with a radius of 1 • Usually with the center on the origin on the coordinate system • Symmetric with respect to the x-axis, y-axis, and the origin

  32. Trigonometric Functions on the Unit Circle • Unit Circle • Consider an angle between 0O and 90O in standard position • Let P(x,y) be where the angle intersects with the unit circle • Draw a perpendicular segment from intersection point back down to the positive x axis • Creates a right triangle • Find the sin Θ and cos Θ • Sin Θ = y; Cos Θ = x

  33. Trigonometric Functions on the Unit Circle • Sine and Cosine on the Unit Circle • If the terminal side of an angle Θ in standard position intersects the unit circle at P(x,y), then cos Θ = x and sin Θ=y.

  34. Trigonometric Functions on the Unit Circle

  35. Trigonometric Functions on the Unit Circle • Circular Functions • Functions defined using the unit circle • Ie Sin and Cosine • How can we define the other cosine functions on the unit circle? • Tan Θ = y/x • Csc Θ =1/y • Sec Θ =1/x • Cot Θ = x/y

  36. Trigonometric Functions on the Unit Circle • Use the unit circle to find each value • Cos (-180O) • Which way do you go for negative angles? • Clockwise • What is the ordered pair of the intersection of this angle on the unit circle? • (-1,0) • Cos Θ = x-axis • Cos (-180O) = -1 • Sec(90O) • Where is the angle located ? • Terminal Side is on positive y axis • Where is the intersection on the unit circle? • Intersection at (0,1) • Sec Θ = 1/x • Sec(90O)=1/0 = undefined

  37. Trigonometric Functions on the Unit Circle • Use the unit circle to find each value • Sin(-90O) • -1 • Cot(270O) • 0

  38. Trigonometric Functions on the Unit Circle • Use the unit circle to find the values of all six trigonometric functions for a 210 degree angle. • What is the intersection with the unit circle? • (-√(3)/2, -1/2) • Sin Θ = -1/2 • Cos Θ =- √(3)/2 • Tan Θ = √(3)/3 • Csc Θ =-2 • Sec Θ =-2 √(3)/3 • Cot Θ = √(3)

  39. Unit Circle Quiz • Use the unit circle to find each value • Tan 3600 • Cos 450 • Sin(-600) • Csc(-2100 ) • Sec(2250 ) • Name the six trigonometry functions for the angles below • 1500 • 4200

  40. Trigonometric Functions on the Unit Circle • What if the angle doesn’t fall within the unit circle? • What if the length is greater or less than 1? • Use length R instead of 1 in unit circle and R = (x2+y2)1/2

  41. Trigonometric Functions on the Unit Circle • What are the six trigonometric values using length r? • Sin Θ =y/r • Cos Θ =x/r • Tan Θ =y/x • Csc Θ =r/y • Sec Θ =r/x • Cot Θ =x/y

  42. Trigonometric Functions on the Unit Circle • Find the six trigonometric functions for angle Θ in standard position if a point with the coordinates (-15, 20) lies on its terminal side. • Draw Figure on xy axis • What are the side measures? • Leg = -3 • Leg = 4 • Hypotenuse = 5 • Where is the theta located? • At the point they gave us • What are the six functions? • Sin Θ =4/5 • Cos Θ =-3/5 • Tan Θ=-4/3 • Csc Θ =5/4 • Sec Θ =-5/3 • Cot Θ =-3/4

  43. Review Quiz 5.3 • Suppose Θ is an angle in standard position whose terminal side lies in Quadrant III. If sin Θ = -4/5, find the values of the remaining five trigonometric functions of Θ? • What do we need to do first? • Draw figure • Next? • Find Missing Side • +/-3, use negative 3 because Quadrant III • Find 5 remaining trig functions. • Cos Θ = -3/5 • Tan Θ =4/3 • Csc Θ =-5/4 • Sec Θ =-5/3 • Cot Θ =3/4

  44. 5.4 Applying Trigonometric Functions • Use trigonometry to find the measures of the sides of right triangles • Make sure calculators are in degrees and not radians • Press Mode • Scroll down to third line and arrow left one • Hit enter • Quit • Quick Check • Cos(90) = 0

  45. Applying Trigonometric Functions • In triangle PRQ, P = 35o and r = 14. Find Q. • First draw figure • How is Q related to Θ? • Adjacent Side • What function should we use? • Cosine • Cos P = q/r • Cos 35=q/14 • 14*cos(35)=Q • Q is about 11.5

  46. Applying Trigonometric Functions • Angles of Elevation • The angle between a horizontal line and the line of sight from an observer to an object at a higher level • Angles of Depression • The angle between a horizontal line and theline of sight from the observer to an object at a lower level

  47. 5.5 Solving Right Triangles Evaluate inverse trig functions. Find missing angle measurements. Solve right triangles.

  48. Solving Right Triangles • Inverse of a Trigonometric Function • The arcsine, arccosine, and arctangent relations with their corresponding trigonometric functions • Arcsine • Sin x = √3/2 • Can be written as x = arcsine √3/2 or x = sin-1 √3/2 • Read this as x is an angle whose sine is √3/2 • Same for other two trig functions • Arccosine • Arctangent

  49. Solving Right Triangles • Sin x = √3/2 • X is an angle with sine √3/2 • X = arcsine √3/2 or x = sin-1 √3/2 • 60 o, 120 o, or any coterminal angles with these • Tan x = 1 • 45 o,225o • Sin x = -1/2 • 210 o, 330 o

  50. Solving Right Triangles • Evaluate and assume angles are in Quadrant I • Cos(arcsin 2/3) • Let B = arcsin 2/3 • Sin B = 2/3 • Draw figure in Quadrant I • Find X • = √5/3 • Try in calculator • Tan(cos-1 4/5) • 3/4

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