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Trigonometry

Trigonometry. Chapters 8.2 - 8.3. 45-45-90 Theorem. 45 0. Hypotenuse. c. Opposite side. b. 45 0. a. Opposite side. 45-45-90 Theorem. The opposite sides of a 45-45-90 triangle are the same length. 45 0.

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Trigonometry

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  1. Trigonometry Chapters 8.2 - 8.3

  2. 45-45-90 Theorem 450 Hypotenuse c Opposite side b 450 a Opposite side

  3. 45-45-90 Theorem • The opposite sides of a 45-45-90 triangle are the same length 450 • Using the Pythagorean theorem we find the hypotenuse is always n times the square root of 2 n 450 n

  4. 45-45-90 Theorem • What is the length of the hypotenuse 450 x 450 x

  5. 45-45-90 Theorem • What is the length of the hypotenuse 450 3 450 3

  6. 45-45-90 Theorem • What is the length of the sides? 450 5 450 5

  7. 45-45-90 Theorem • What is the length of the sides? • Remember, the hypotenuse is 2 times a side 450 x 450 6 = x * 2 x

  8. Divide by 2 • Rationalize the Denominator 6 = x * 2 2 2 6 = x 2

  9. 2 62 6 * = 32 = 2 2 2 * Multiply top and bottom by 2 Simplify the fraction

  10. 30-60-90 Theorem Hypotenuse 600 Opposite of 30 300 Opposite of 60

  11. 30-60-90 Theorem • The opposite of the 300 angle is n Hypotenuse 600 Opposite of 30 n 300 Opposite of 60

  12. 30-60-90 Theorem • The opposite of the 600 angle is n3 Hypotenuse 600 n 300 Opposite of 60 n3

  13. 30-60-90 Theorem • The opposite of the right angle is 2n 2n Hypotenuse 600 n 300 n3

  14. 30-60-90 Theorem • Find the lengths of the other two sides 2x 600 x 300 x3

  15. 30-60-90 Theorem • Find the lengths of the other two sides 2(4) 8 600 4 300 43

  16. 30-60-90 Theorem • Find the lengths of the other two sides 2(5) 10 600 5 300 53

  17. 30-60-90 Theorem • Find the lengths of the other two sides 20 600 10 300 103

  18. 30-60-90 Theorem • Find the lengths of the other two sides 600 300 8 • First find the length of side opposite the 30

  19. 30-60-90 Theorem • Call the side x 600 x 300 8 • x times 3 = 8

  20. x3 = 8 3 3 8 3 83 * x = = 3 3 3 *

  21. 30-60-90 Theorem • Hypotenuse is 2 times the side opposite the 300 angle 163 83 600 3 3 300 8

  22. Trigonometry • Trigonometric Ratios- • Similar right triangles have equivalent ratios for its corresponding sides a c b = = e f d d a b e f c

  23. Sine Length of opposite side • Sine of óB = Length of hypotenuse Sine (sin) of an angle = ratio of opposite side over the hypotenuse C a b B A c

  24. Sine Length of opposite side • Sine of óB = Length of hypotenuse C Hypotenuse a b Opposite B A c adjacent

  25. Sine b opp = • Sin B = a hyp C Hypotenuse a b Opposite B A c adjacent

  26. Cosine adjacent side • Cosine of óB = hypotenuse Cosine (cos) of an angle = ratio of adjacent side over the hypotenuse C a b B A c

  27. Cosine adjacent side • Cosine of óB = hypotenuse C Hypotenuse a b Opposite B A c adjacent

  28. Cosine c adj = • Cos B = a hyp C Hypotenuse a b Opposite B A c adjacent

  29. Tangent opposite side • Tangent of óB = adjacent side Tangent (tan) of an angle = ratio of opposite side over the adjacent side C a b B A c

  30. Tangent opposite side • Tangent of óB = adjacent side C Hypotenuse a b Opposite B A c adjacent

  31. Tangent b opp = • Tan B = c adj C Hypotenuse a b Opposite B A c adjacent

  32. Trigonometry • How to remember the order: SohCahToa opp • Sin x = hyp opp • Tan x = adj adj • Cos x = hyp

  33. Trigonometry • Find the sine, cosine, and tangent ratios of óB C Hypotenuse 17 8 Opposite B A 15 adjacent

  34. opp 8 • Sin B = 17 hyp opp 8 • Tan B = adj 15 15 adj • Cos B = 17 hyp C Hypotenuse 17 8 Opposite B A 15 adjacent

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