The TrigonometricComparations & Function
Trigonometric Ratios 1. Trigonometric Ratios Formed by an Angle     of a Right Triangle                   a² + b² = c²    ...
a. Understanding of Sine (sin), Cosine (cos),    and Tangent (tan) sin α = opposite side / hypotenuse = a / ccos α = adjac...
Example 1:Determine the value of sine, cosine andtangent of BAC and         ABC in the trianglebelow, if a = 6 and b = 8.A...
then sin   BAC = sin α = a/c = 6/10     cos   BAC = cos α = b/c = 8/10     tan   BAC = tan α = a/b = 6/8    sin    ABC = s...
Note:In addition has the reverse function thetrigonometry ratios also have conclution.If α + β = 90°, then sin α = cos β.E...
b. The Value of Trigonometric Ratios for   Specific AnglesFor Angle 45°Picture 2. Trigonometric ratios of angle 45 °      ...
Table 1. Trigonometric Ratios of Specific                      Angle α        0°     30°     45°     60°     90°sin α     ...
2. Trigonometric Ratios Formula of Related   Angles    a. Trigonometric Ratios in Quadrant I    Y                 B’ (x’ ,...
The value of trigonometry ratios for angle        (90° - α) is the following:     sin (90° - α) = y’/r = x/r = cos α     c...
b. The Trigonometric Ratios in Quadrant II                Picture 4. The Trigonometric Ratios                             ...
The trigonometric ratios value of angle         (180° - α) is as follow: sin (180° - α) = y’/r = y/r = sin α cos (180° - α...
c. Trigonometric Ratios in Quadrant III                    Picture 5. Trigonometric Ratios                                ...
The trigonometric ratios value of angle         (180° + α) is as follow:sin (180° + α) = y’/r    = - y/r   = sin αcsc (180...
d. Trigonometric Ratios in Quadrant IV                                                       Picture 6.             Y     ...
The trigonometric ratios value of angle         (360° - α) is as follow:sin (360° - α) = y’/r = - y/r = - sin αcsc (360° -...
Trigonometric Identity      There are four trigonometric identities     formula that you should know, as follow:a.   tan α...
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Matematika - The Trigonometric Comparations & Function

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Matematika - The Trigonometric Comparations & Function

  1. 1. The TrigonometricComparations & Function
  2. 2. Trigonometric Ratios 1. Trigonometric Ratios Formed by an Angle of a Right Triangle a² + b² = c² Picture 1. BA right triangle ABC hypotenuse β c a right angle side ) α _ A l C b right angle side
  3. 3. a. Understanding of Sine (sin), Cosine (cos), and Tangent (tan) sin α = opposite side / hypotenuse = a / ccos α = adjacent side / hypotenuse = b / ctan α = opposite side / adjacent side = a / b
  4. 4. Example 1:Determine the value of sine, cosine andtangent of BAC and ABC in the trianglebelow, if a = 6 and b = 8.Answers:From the figure, known that AC = b = 8,BC = a = 6, and AB = c. The value c can becalculated by using ‘Pythagoras theorem’.c² = a² + b²c² = 6² + 8² = 100c = √100 = 10
  5. 5. then sin BAC = sin α = a/c = 6/10 cos BAC = cos α = b/c = 8/10 tan BAC = tan α = a/b = 6/8 sin ABC = sin β = b/c = 8/10 cos ABC = cos β = a/c = 6/10 tan ABC = tan β = b/a = 8/6 A α c 8 _ B ) β 6 l C
  6. 6. Note:In addition has the reverse function thetrigonometry ratios also have conclution.If α + β = 90°, then sin α = cos β.Example:sin 30° = cos 60°,sin 40° = cos 50°,sin 20° = cos 70° and so on.
  7. 7. b. The Value of Trigonometric Ratios for Specific AnglesFor Angle 45°Picture 2. Trigonometric ratios of angle 45 ° C D C 45° d √2 1 1 A 45° B A 45° B 1 1
  8. 8. Table 1. Trigonometric Ratios of Specific Angle α 0° 30° 45° 60° 90°sin α 0 1/2 1/2√2 1/2√3 1cos α 1 1/2√3 1/2√2 1/2 0tan α 0 1/3√3 1 √3 ∞
  9. 9. 2. Trigonometric Ratios Formula of Related Angles a. Trigonometric Ratios in Quadrant I Y B’ (x’ , y’) Picture 3. x’ y=x Angle (90° - α) y’ B(x , y) in Quadrant I r α r y α X0 x
  10. 10. The value of trigonometry ratios for angle (90° - α) is the following: sin (90° - α) = y’/r = x/r = cos α cos (90° - α) = x’/r = y/r = sin α tan (90° - α) = y’/x’ = x/y = cot α csc (90° - α) = r/y’ = r/x = sec α sec (90° - α) = r/x’ = r/y = csc α cot (90° - α) = x’/y’ = y/x = tan α
  11. 11. b. The Trigonometric Ratios in Quadrant II Picture 4. The Trigonometric Ratios in Quadrant II Y B’ (x’ , y) B (x , y) r (180° - α) r y’ y x’ x X α α 0
  12. 12. The trigonometric ratios value of angle (180° - α) is as follow: sin (180° - α) = y’/r = y/r = sin α cos (180° - α) = x’/r = - x/r = - cos α tan (180° - α) = y’/x’ = - y/x = - tanα csc (180° - α) = r/y’ = r/y = csc α sec (180° - α) = r/x’ = - r/x = - sec α cot (180° - α) = x’/y’ = - x/y = - cot α
  13. 13. c. Trigonometric Ratios in Quadrant III Picture 5. Trigonometric Ratios in Quadrant III Y B (x , y) (180° + α) y x’ α x α X 0 y’B’ (x’ , y)
  14. 14. The trigonometric ratios value of angle (180° + α) is as follow:sin (180° + α) = y’/r = - y/r = sin αcsc (180° +α) = r/y’ = - r/y = - csc αcos (180° + α) = x’/r = - x/r = - cos αsec (180° +α) = r/x’ = - r/x = - sec αtan (180° + α) = y’/x’ = y/x = tan αcot (180° + α) = x’/y’ = x/y = cot α
  15. 15. d. Trigonometric Ratios in Quadrant IV Picture 6. Y B (x , y) Trigonometric Ratios r y in Quadran IV(360° - α) α 0 α X r B’ (x’ , y’)
  16. 16. The trigonometric ratios value of angle (360° - α) is as follow:sin (360° - α) = y’/r = - y/r = - sin αcsc (360° -α) = r/y’ = - r/y = - csc α cos (360° - α) = x’/r = x/r = cos α sec (360° -α) = r/x’ = r/x = sec αtan (360° - α) = y’/x’ = - y/x = - tan αcot (360° - α) = x’/y’ = - x/y = - cot α
  17. 17. Trigonometric Identity There are four trigonometric identities formula that you should know, as follow:a. tan α = sin α / cos α , cot α = cos α / sin αb. sin² α + cos² α = 1c. 1 + tan² α = sec² αd. 1 + cot² α = csc² α
  18. 18. THANK YOU

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