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Pre Calculus

Pre Calculus . Day 37. Plan for the Day. Section 4.1 Intro to Trigonometry Trigonometry - “Measurement of Triangles” Key Terms Radians, Degrees and Conversions The Calculator Co-terminal Angles Arc Measures Short Quizzes Every Day. What You Should Learn. Describe angles.

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Pre Calculus

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  1. Pre Calculus Day 37

  2. Plan for the Day • Section 4.1 Intro to Trigonometry • Trigonometry - “Measurement of Triangles” • Key Terms • Radians, Degrees and Conversions • The Calculator • Co-terminal Angles • Arc Measures • Short Quizzes Every Day

  3. What You Should Learn • Describe angles. • Use radian measure. • Use degree measure. • Convert between radians and degrees • Use angles to model and solve real-life problems.

  4. Angles, ,  or A, B, C • Angle – Rotation of a ray about its endpoint • Initial Side – The starting position of the ray • Terminal Side – The position of the ray after rotation • Standard Position – An angle placed on the coordinate plane with the initial side coincides with the positive x-axis • Positive angles – generated by counterclockwise rotation • Negative angles – generated by clockwise rotation • Co terminal – angles with the same initial and terminal sides • Measure of an Angle – The amount of rotation from the initial side to the terminal side • Central Angle – An angle whose vertex is the center of a circle

  5. Angles • As derived from the Greek language, the word trigonometry means “measurement of triangles.” • Initially, trigonometry dealt with relationships among the sides and angles of triangles. • An angle is determined by rotating a ray (half-line) about its endpoint.

  6. C A θ Initial Side Terminal Side B Vertex Angles An angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side.

  7. Angles • The endpoint of the ray is the vertex of the angle. • This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x-axis. • Such an angle is in standard position. Angle in standard position

  8. Degree Measure • One way to measure angles is in terms of degrees, denoted by the symbol . • A measure of one degree (1) is equivalent to a rotation of of a complete revolution about the vertex. • To measure angles, it is convenient to mark degrees on the circumference of a circle. Figure 4.13

  9. 180º  90º  Acute angle 0º <  < 90º Right angle 1/4 rotation Obtuse angle 90º <  < 180º Straight angle 1/2 rotation Measuring Angles Using Degrees The figures below show angles classified by their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.

  10. y y is positive Terminal Side  Vertex Initial Side x x Vertex  Terminal Side Initial Side is negative Positive angles rotate counterclockwise. Negative angles rotate clockwise. Angles of the Rectangular Coordinate System • An angle is in standard position if • its vertex is at the origin of a rectangular coordinate system and • its initial side lies along the positive x-axis.

  11. When an angle is in standard position, it is important to know what quadrant your angle has its terminal side • What are the quadrants??

  12. Circle with Degrees 90o Quadrant I 0o<< 90o Quadrant II 90o<< 180o 180o 0o 360o Quadrant III 180o<< 270o Quadrant IV 270o<< 360o 270o

  13. Positive Angles 0o<< 90o Quadrant I (acute)  = 90o Y-axis (right) 90o<<180o Quadrant II (obtuse)  = 180o X-axis (straight) 180o<<270o Quadrant III  = 270o Y-axis 270o<<360o Quadrant IV  = 360o X-axis Negative Angles 0o>> -90o Quadrant IV  = -90o Y-axis -90o>>-180o Quadrant III  = -180o X-axis -180o>>-270o Quadrant II  = -270o Y-axis -270o>>-360o Quadrant I  = -360o X-axis Angle Measures - Degrees

  14. What quadrant is it in? a) 30o b) 120o c) -45o d) 190o e) 370o f) -95o g) 280o h) 820o i) -745o

  15. Coterminal Angles Angle A Angle B Angle A and B are coterminal

  16. An angle of xº is coterminal with angles of xº± k· 360º where k is an integer. Coterminal Angles

  17. Co-terminal θ + 360o θ – 360o Name two co terminal angles, one positive and one negative • 30o 30o + 360o = 390o 30o - 360o = -330o • 120o 120o + 360o = 480o 120o - 360o = -240o • -45o -45o + 360o = 315o -45o - 360o = -405o

  18. Degrees, Minutes, Seconds Measurement in Degrees A minute is 1/60 of a degree – denoted with ’ A second is 1/60 of a minute – denoted with ” Ex: 150o 35’ 25” Where is it on the calculator?? Converting from decimal form to DMS

  19. Radian Measure • A second way to measure angles is in radians. This type of measure is especially useful in calculus. • To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle. Arc length = radius when  =1 radian Figure 4.5

  20. Radian - Definition One radian is the angle subtended at the center of a circle by an arc of circumference that is equal in length to the radius of the circle. In terms of a circle it can be seen as the ratio of the length of the arc subtended by two radii to the radius of the circle. http://en.wikipedia.org

  21. How many Radians are in a circle? Because the circumference of a circle is 2r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of s = 2r. If a circle has a radius of 1 what is its circumference? Circumference = πd or 2πr Circumference = 2π(1) or 2π There are 2π radians in every circle • How about a half circle? A quarter circle?

  22. Radian Measure • Because the units of measure for s and r are the same, the ratio s/r has no units—it is simply a real number. • Moreover, because 26.28, there are just over six radius lengths in a full circle. Figure 4.6

  23. Radian Measure Because the radian measure of an angle of one full revolution is 2, you can obtain the following.

  24. Radian Measure • These and other common angles we will frequently use.

  25. Circle with Degrees and Radians 90o or π/2 180o or π 0o or 0 360o or 2π 270o or 3π/2

  26. Positive Angles 0<<π/2 Quadrant I (acute) =π/2 Y-axis (right) π/2<< Quadrant II (obtuse)  =  X-axis (straight) <<3π/2 Quadrant III =3π/2 Y-axis 3π/2<<2 Quadrant IV =2 X-axis Negative Angles 0>>-π/2 Quadrant IV =-π/2 Y-axis -π/2>>- Quadrant III  = - X-axis ->>-3π/2 Quadrant II =-3π/2 Y-axis -3π/2>>-2 Quadrant I =-2 X-axis Angle Measures - Radians One radian is the measure of a central angle  that intercepts an arc s equal in length of the radius r of the circle.

  27. What quadrant is it in? a) π/3b) π/12c) -π/3 d) 5π/6e) 3π/4f) -4π/3 g) π/6h) -5π/6i) 7π/3

  28. Radian Measure • Two angles are coterminal if they have the same initial and terminal sides. For instance, the angles 0 and 2 are coterminal, as are the angles  / 6 and 13 / 6. • You can find an angle that is coterminal to a given angle  by adding or subtracting 2 (one revolution). • A given angle  has infinitely many coterminal angles. For instance,  =  / 6 is coterminal where n is an integer.

  29. An angle of x is coterminal with angles of x ± k· 2π where k is an integer. Coterminal Angles

  30. Co-terminal θ + 2π θ – 2π Name two coterminal angles, one positive and one negative a) π/2b) π/12c) -π/4

  31. Calculator Issues • Mode – you must know what you are working with!!! • Xo means degrees • X with no symbol is radians – it is the default! • Can be written Xr

  32. Other Angle Properties • Two positive angles  and  are complementary (complements of each other) if their sum is  / 2 (radians) or 90o (degrees). • Two positive angles are supplementary (supplements of each other) if their sum is  (radians) or 180o (degrees). Supplementary angles Complementary angles

  33. Finding Complements and Supplements • For an xº angle, the complement is a 90º – xº or  / 2 – xº angle. • Thus, the complement’s measure is found by subtracting the angle’s measure from 90º or  / 2. • For an xº angle, the supplement is a 180º – xº or  – xº angle. • Thus, the supplement’s measure is found by subtracting the angle’s measure from 180º or  .

  34. Find the Complement and Supplement • π/3 • π/12 • 55o • 5π/6 • 75o

  35. Degree Measure • When no units of angle measure are specified, radian measure is implied. • For instance, if you write  = 2, you imply that  = 2 radians. • Most radian measures you work with will have π in them.

  36. Conversion between Degrees and Radians • Using the basic relationship  radians = 180º, • To convert degrees to radians, multiply degrees by ( radians) / 180 • To convert radians to degrees, multiply radians by 180 / ( radians)

  37. Convert to the other form … • π/3 • π/12 • 55o • -5π/6 • -75o

  38. Let r be the radius of a circle and  the non- negative radian measure of a central angle of the circle. The length of the arc intercepted by the central angle is s = r  s  O r Length of a Circular Arc

  39. Finding Arc Length • A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240.

  40. Solution • First, we convertto radians • Then, using a radius of r = 4 inches, you can find the arc length to bes = r

  41. Try these: • Find the length of the arc of a circle with a radius of 5 and a central angle of 5 radians. • Find the length of the arc of a circle with a radius of 3 and a central angle of π/4 radians. • Find the length of the radius of a circle with an arc length of 10 and a central angle of π/6 radians. • Find the length of the arc of a circle with a radius of 7 and a central angle of 50o.

  42. Homework 19 • Section 4.1Page 269 -271 # 7, 9, 17, 20, 21, 23, 27, 31, 38, 39, 43, 47, 56, 57, 63, 66, 71, 77, 81, 85, 89, 91

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