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Gettin’ Triggy wit it

Learn about angles, their measure, and rotation direction. Discover how to convert between degrees and radians, calculate arc length, and work with angle sectors in this comprehensive tutorial.

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Gettin’ Triggy wit it

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  1. Gettin’ Triggy wit it http://www.youtube.com/watch?v=t2uPYYLH4Zo

  2. 6.1 Angles and Their Measure

  3. An angle is formed by joining the endpoints of two half-lines called rays. The side you measure to is called the terminal side. Angles measured counterclockwise are given a positive sign and angles measured clockwise are given a negative sign. Terminal Side Positive Angle This is a counterclockwise rotation. Negative Angle This is a clockwise rotation. Initial Side The side you measure from is called the initial side.

  4. It’s Greek To Me! It is customary to use small letters in the Greek alphabet to symbolize angle measurement.    alpha beta gamma  theta

  5. We can use a coordinate system with angles by putting the initial side along the positive x-axis with the vertex at the origin. Quadrant IIangle Quadrant Iangle Terminal Side  positive  negative Initial Side Quadrant IVangle If the terminal side is along an axis it is called a quadrantal angle. We say the angle lies in whatever quadrant the terminal side lies in.

  6. We will be using two different units of measure when talking about angles: Degrees and Radians If we start with the initial side and go all of the way around in a counterclockwise direction we have 360 degrees  = 360°  = 90° If we went 1/4 of the way in a clockwise direction the angle would measure -90° You are probably already familiar with a right angle that measures 1/4 of the way around or 90°  = - 90° Let’s talk about degrees first. You are probably already somewhat familiar with degrees.

  7. What is the measure of this angle? You could measure in the positive direction and go around another rotation which would be another 360°  = - 360° + 45°  = - 315°  = 45° You could measure in the positive direction  = 360° + 45° = 405° You could measure in the negative direction There are many ways to express the given angle. Whichever way you express it, it is still a Quadrant I angle since the terminal side is in Quadrant I.

  8. If the angle is not exactly to the next degree it can be expressed as a decimal (most common in math) or in degrees, minutes and seconds (common in surveying and some navigation). 1 degree = 60 minutes 1 minute = 60 seconds  = 25°48‘29" degrees seconds minutes To convert to decimal form use conversion fractions. These are fractions where the numerator = denominator but two different units. Put unit on top you want to convert to and put unit on bottom you want to get rid of. Let's convert the seconds to minutes 30" = 0.5'

  9. 1 degree = 60 minutes 1 minute = 60 seconds  = 25°48'30" = 25°48.5' = 25.808° Now let's use another conversion fraction to get rid of minutes. 48.5' = .808°

  10. Another way to measure angles is using what is called radians. Given a circle of radius r with the vertex of an angle as the center of the circle, if the arc length formed by intercepting the circle with the sides of the angle is the same length as the radius r, the angle measures one radian. terminal side arc length is also r r r r initial side This angle measures 1 radian radius of circle is r

  11. 3 0.52 Arc length s of a circle is found with the following formula: IMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA! s = r arc length radius measure of angle Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian. arc length to find is in black  = 0.52 s = r 3 = 1.56 m What if we have the measure of the angle in degrees? We can't use the formula until we convert to radians, but how?

  12. We need a conversion from degrees to radians. We could use a conversion fraction if we knew how many degrees equaled how many radians. If we look at one revolution around the circle, the arc length would be the circumference. Recall that circumference of a circle is 2r s = r Let's start with the arc length formula 2r = r cancel the r's 2 =  This tells us that the radian measure all the way around is 2. All the way around in degrees is 360°. 2  radians = 360°

  13. = radians 2  radians = 360°  radians = 180° Convert 30° to radians using a conversion fraction. The fraction can be reduced by 2. This would be a simpler conversion fraction. 30° 180° Can leave with  or use  button on your calculator for decimal.  0.52 Convert /3 radians to degrees using a conversion fraction. = 60°

  14. Area of a Sector of a Circle The formula for the area of a sector of a circle (shown in red here) is derived in your textbook. It is:  r Again  must be in RADIANS so if it is in degrees you must convert to radians to use the formula. Find the area of the sector if the radius is 3 feet and  = 50° = 0.873 radians

  15. A Sense of Angle Sizes See if you can guess the size of these angles first in degrees and then in radians. You will be working so much with these angles, you should know them in both degrees and radians.

  16. Angles, Arc length, Conversions Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative. Coterminal Angles: Angles that have the same terminal side. 60°, 420°, and –300° are all coterminal. Degrees to radians: Multiply angle by radians Radians to degrees: Multiply angle by Note: 1 revolution = 360° = 2π radians. Arc length = central angle x radius, or Note: The central angle must be in radian measure.

  17. 6.2 TRIGONOMETRY

  18. Remember SOHCAHTOA Sine is Opposite divided by Hypotenuse Cosine is Adjacent divided by Hypotenuse Tangent is Opposite divided by Adjacent SOHCAHTOA!!!!!!

  19. http://www.youtube.com/watch?v=4iC-gjKvc7A&feature=related

  20. Right Triangle Trig Definitions sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a B c a A C b

  21. Special Right Triangles 30° 45° 2 1 60° 45° 1 1

  22. 6.2 Assignment (day 1) pp. 417-419 (1-23 odd, 27, 29) 21, 23

  23. Basic Trigonometric Identities Quotient identities: Even/Odd identities: Even functions Odd functions Odd functions Reciprocal Identities: Pythagorean Identities:

  24. All Students Take Calculus. Quad II Quad I cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0 Quad IV Quad III

  25. http://www.youtube.com/watch?v=Ah9NxAGzlDc&feature=related

  26. 6.3 Trigonometric Functions of Real Numbers

  27. Exact Values Using Points on the Circle • Example. Let t be a real number and P = the point on the unit circle that corresponds to t. Problem: Find the values of sin t, cos t, tan t, csc t, sec t and cot t Answer:

  28. Exact Values for Quadrantal Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = 0 = 0± Answer:

  29. Exact Values for Quadrantal Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = = 90± Answer:

  30. Exact Values for Quadrantal Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = ¼ = 180± Answer:

  31. Circles of Radius r • Example. Problem: Find the exact values of each of the trigonometric functions of an angle θ if ({12, 5}) is a point on its terminal side. Answer:

  32. Even-Odd Properties • Theorem. [Even-Odd Properties] sin(θ) = sin(θ) cos(θ) = cos(θ) tan(θ) = tan(θ) csc(θ) = csc(θ) sec(θ) = sec(θ) cot(θ) = cot(θ) • Cosine and secant are even functions • The other functions are odd functions

  33. The Sine Function The sine of a real number t is the y–coordinate (height) of the point P in the following diagram, where |t| is the length of the arc. y 1 P sin t |t| –1 x 1 1 unit –1

  34. 2. The range is the set of y values such that . 5. Each function cycles through all the values of the range over an x-interval of . Properties of Sine and Cosine Functions Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of real numbers. 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 6. The cycle repeats itself indefinitely in both directions of thex-axis.

  35. 1.5 1 0.5 0 -1.5 -1 -0.5 0.5 1 1.5 -0.5 -1 -1.5 The Sine Function Highlight those sections of the circle where sin(t) >0 sin(t) > 0

  36. The Sine Function

  37. x 0 sin x 0 1 0 -1 0 y = sin x y x Graph of the Sine Function Sine Function To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts. Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

  38. The Cosine Function The cosine of a real number t is the x–coordinate (length) of the point P in the following diagram, where |t| is the length of the arc. y 1 P |t| 1 unit –1 x 1 cos t –1

  39. 1.5 1 0.5 0 -1.5 -1 -0.5 0.5 1 1.5 -0.5 -1 -1.5 The Cosine Function Highlight those sections where cos(t) > 0 cos(t) > 0

  40. The Cosine Function

  41. x 0 cos x 1 0 -1 0 1 y = cos x y x Graph of the Cosine Function Cosine Function To sketch the graph of y = cos x first locate the key points.These are the maximum points, the minimum points, and the intercepts. Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

  42. Graphing the Tangent Function • Periodicity: Only need to graph on interval [0, ¼] • Plot points and graph

  43. Graphing the Cotangent Function • Periodicity: Only need to graph on interval [0, ¼]

  44. Graphing the Cosecant and Secant Functions • Use reciprocal identities • Graph of y = csc x

  45. Graphing the Cosecant and Secant Functions • Use reciprocal identities • Graph of y = sec x

  46. 6.4 Values of the Trigonometric Functions

  47. 6.4 Reference Angles • A reference angle for an angle , written  , is the positive acute angle made by the terminal side of angle  and the x-axis. Example Find the reference angle for each angle. • 218º (b) Solution (a)   = 218º – 180º = 38º (b)

  48. 6.4 Special Angles as Reference Angles Example Find the values of the trigonometric functions for 210º. Solution The reference angle for 210º is 210º – 180º = 30º. Choose point P on the terminal side so that the distance from the origin to P is 2. A 30º - 60º right triangle is formed.

  49. 6.4 Finding Trigonometric Function Values Using Reference Angles Example Find the exact value of each expression. • cos(–240º) (b) tan 675º Solution • –240º is coterminal with 120º. The reference angle is 180º – 120º = 60º. Since –240º lies in quadrant II, the cos(–240º) is negative. • Similarly, tan 675º = tan 315º = –tan 45º = –1.

  50. Reference Angles Quad I Quad II θ’ = 180° – θ θ’ = θ θ’ = π – θ θ’ = θ– 180° θ’ = 360° – θ θ’ = θ– π θ’ = 2π – θ Quad III Quad IV

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