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2.1 Trigonometry

2.1 Trigonometry. Vocabulary:. Angle – created by rotating a ray about its endpoint. Initial Side – the starting position of the ray. Terminal Side – the position of the ray after rotation. Vertex – the endpoint of the ray.

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2.1 Trigonometry

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  1. 2.1 Trigonometry

  2. Vocabulary: Angle – created by rotating a ray about its endpoint. Initial Side – the starting position of the ray. Terminal Side – the position of the ray after rotation. Vertex – the endpoint of the ray.

  3. This arrow means that the rotation was in a counterclockwise direction. Terminal side Vertex Initial side This arrow means that the rotation was in a clockwise direction. Initial side Vertex Terminal side

  4. Positive Angles – angles generated by a counterclockwise rotation. Negative Angles – angles generated by a clockwise rotation. We label angles in trigonometry by using the Greek alphabet.  - Greek letter alpha  - Greek letter beta  - Greek letter phi  - Greek letter theta

  5. This represents a positive angle Terminal side Vertex Initial side This represents a negative angle Initial side Vertex Terminal side

  6. Standard Position – an angle is in standard position when its initial side rests on the positive half of the x-axis. Positive angle in standard position

  7. There are two ways to measure angles… Degrees Radians

  8. Degrees: • There are 360 in a complete circle. • 1 is 1/360th of a rotation. • Radians: • There are 2 radians in a complete circle. • 1 radian is the size of the central angle when the • radius of the circle is the same size as the arc of • the central angle.

  9. Length of the arc is equal to the length of the radius. arc 1 Radian radius

  10. Coterminal angles – two angles that share a common vertex, a common initial side and a common terminal side. Examples of Coterminal Angles  and  are coterminal angles because they share the same initial side and same terminal side.   Coterminal angles could go in opposite directions.

  11. Examples of Coterminal Angles  and  are coterminal angles because they share the same initial side and same terminal side.   Coterminal angles could go in the same direction with multiple rotations.

  12. Finding coterminal anglesof angles measured in degrees: Since a complete circle has a total of 360, you can find coterminal angles by adding or subtracting 360 from the angle that is provided.

  13. Example: Find two coterminal angles (one positive and one negative) for the following angles.  = 25 positive coterminal angle: 25 + 360 = 385 negative coterminal angle: 25 – 360 = - 335

  14. Example: Find two coterminal angles (one positive and one negative) for the following angles.  = 725 positive coterminal angle: 725 + 360 = 1085 (add a rotation) or 725 – 360 = 365 (subtract a rotation) or 725 – 360 – 360 = 5 (subtract 2 rotations) negative coterminal angle: 725 – 360 – 360 – 360 = - 355 (must subtract 3 rotations)

  15. Example: Find two coterminal angles (one positive and one negative) for the following angles.  = -90 positive coterminal angle: -90 + 360 = 270 negative coterminal angle: - 90 – 360 = - 470

  16. Finding coterminal angles of angles measured in radians: Since a complete circle has a total of 2 radians you can find coterminal angles by adding or subtracting 2 from the angle that is provided.

  17. Example: Find two coterminal angles (one positive and one negative) for the following angles.  = /7 positive coterminal angle: /7 + 2 = /7 + 14/7 = 15/7 rad negative coterminal angle: /7 - 2 = /7 - 14/7 = -13/7 rad

  18. Example: Find two coterminal angles (one positive and one negative) for the following angles.  = -4/9 positive coterminal angle: -4/9 +2 = -4/9 + 18/9 =14/9 rad negative coterminal angle: -4/9 -2 =-4/9 - 18/9 =-22/9 rad

  19. Complementary angles – twopositive angles whose sum is 90 or two positive angles whose sum is /2. To find the complement of a given angle you subtract the given angle from 90 (if the angle provided is in degrees) or from /2 (if the angle provided is in radians).

  20. . Example: Find the complement of the following angles if one exists.  = 29 complement = 90 – 29 = 61  = 107 complement = 90 – 107 = none (No complement because it is negative)  = /5 complement = /2 - /5 = 5/10 - 2/10 = 3/10

  21. Supplementary angles – twopositive angles whose sum is 180 or two positive angles whose sum is . To find the supplement of a given angle you subtract the given angle from 180 (if the angle provided is in degrees) or from  (if the angle provided is in radians).

  22. Example: Find the supplement of the following angles if one exists.  = 29 supplement = 180 – 29 = 151  = 107 supplement = 180 – 107 = 73  = /5 supplement = - /5 = 5/5 - /5 = 4/

  23. We have to become comfortable working with both forms of measuring angles. Therefore, MEMORIZE the following: We will memorize more, very, very soon.

  24. Manually Converting from Degrees to Radians: • Multiply the given degrees by  radians/180 Example: Convert the following degrees to radians 135 degrees  radians = 1 180 degrees 135 135 radians = 180 3radians 4

  25. Multiply the given degrees by  radians/180 Example: Convert the following degrees to radians 540 degrees  radians = 1 180 degrees 540 540 radians = 180 3radians 1

  26. Manually Converting from Radians to Degrees: • Multiply the given radians by 180/ radians Example: Convert the following radians to degrees. - radians 180 degrees = 3  radians -/3 radians -180 degrees = 3 -60

  27. Multiply the given radians by 180/ radians Example: Convert the following radians to degrees. 9 radians 180 degrees = 2  radians 9/2 radians 1620 degrees = 2 810

  28. Multiply the given radians by 180/ radians Example: Convert the following radians to degrees. 2 radians 180 degrees = 1  radians 2 (if you don’t see the degree symbol, then the angle measure is automatically believed to be a radian.) 360 degrees = 2 114.59

  29. Tomorrow, we will look at your individual calculators and show you how to do these conversions via those calculators. BRING YOUR OWN SCIENTIFIC CALCULATOR TOMORROW!

  30. Finding Arc Length: • The following formula is used to determine arc length: • s = r  Measure of the central angle in radians. arc length radius must have the same units of measure

  31. Examples s = ? 3 radians r= 14 inches s = r  s = (14)(3) s = 42 inches Picture not drawn to scale.

  32. Examples s =9 cm 30 You must convert 30 to radians. r= ? s = r  9 = (r)(/6) r = 54/  cm  17.19 cm Picture not drawn to scale.

  33. Assignment: pg. 91: 1-28, 43-50

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