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4-3 Right Triangle Trigonometry

4-3 Right Triangle Trigonometry. Pre-Calculus. θ. The six trigonometric functions of a right triangle, with an acute angle  , are defined by ratios of two sides of the triangle. hyp. opp. The sides of the right triangle are:. adj.  the side opposite the acute angle  ,.

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4-3 Right Triangle Trigonometry

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  1. 4-3 Right Triangle Trigonometry Pre-Calculus

  2. θ The six trigonometric functions of a right triangle, with an acute angle ,are defined by ratios of two sides of the triangle. hyp opp The sides of the right triangle are: adj  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle.

  3. Trigonometric Functions θ adj opp sin  = cos  = tan  = hyp adj hyp The trigonometric functions are opp adj sine, cosine, tangent, cotangent, secant, and cosecant.

  4. Draw a 30-60-90 triangle using a protractor, you can choose your own value of x. x

  5. Draw a 45-45-90 triangle using a protractor (or properties of isosceles triangles), you can choose your own value of x. x Repeat for at least one additional value of x for each triangle. Then determine the relationships between the sides using the x as a variable to represent any length.

  6. Special Right Triangles

  7. Some basic trig values

  8. Trigonometric Functions θ adj opp sin  = cos  = tan  = csc  = sec  = cot  = hyp adj hyp hyp adj opp adj opp hyp The trigonometric functions are opp adj sine, cosine, tangent, cotangent, secant, and cosecant. Note: sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals.

  9. Reciprocal Functions Another way to look at it… sin  = 1/csc csc = 1/sin cos = 1/sec sec = 1/cos tan = 1/cot cot = 1/tan

  10. 12  5 Given 2 sides of a right triangle you should be able to find the value of all 6 trigonometric functions. Example:

  11. Example: Six Trig Ratios 5 4  3 sin  = sin α = cos  = cos α = tan  = cot  = cot α = tan α = sec  = sec α = csc  = csc α = Calculate the trigonometric functions for  . Calculate the trigonometric functions for . The six trig ratios are What is the relationship of α and θ? They are complementary (α = 90 – θ)

  12. Cofunctions sin  = cos (90  ) cos  = sin (90  ) sin  = cos (π/2  ) cos  = sin(π/2  ) tan  = cot (90  ) cot  = tan (90  ) tan  = cot(π/2  ) cot  = tan(π/2  ) sec  = csc (90  ) csc  = sec (90  ) sec  = csc(π/2  ) csc  = sec(π/2  )

  13. Example: Using Trigonometric Identities Trigonometric Identities are trigonometric equations that hold for all values of the variables. We will learn many Trigonometric Identities and use them to simplify and solve problems.

  14. hyp opp adj θ adj opp hyp adj Quotient Identities sin  = cos  = tan  = The same argument can be made for cot… since it is the reciprocal function of tan.

  15. Quotient Identities Important Question: Why do mathematicians never go the beach?

  16. Pythagorean Identities Three additional identities that we will use are those related to the Pythagorean Theorem: Pythagorean Identities Hmm, what equations 1 can i create? 1 y = sin x = cos 

  17. Pythagorean Identities Three additional identities that we will use are those related to the Pythagorean Theorem: Pythagorean Identities sin2  + cos2  = 1 tan2 + 1 = sec2  cot2  + 1 = csc2 

  18. Identities we have reviewed so far…

  19. Fundamental Trigonometric Identities for Fundamental Trigonometric Identities Reciprocal Identities sin  = 1/csc cos = 1/sec tan = 1/cotcot = 1/tan sec = 1/cos csc = 1/sin Co function Identities sin  = cos(90  ) cos  = sin(90  ) sin  = cos (π/2  ) cos  = sin(π/2  ) tan  = cot(90  ) cot  = tan(90  ) tan  = cot(π/2  ) cot  = tan(π/2  ) sec  = csc(90  ) csc  = sec(90  ) sec  = csc(π/2  ) csc  = sec(π/2  ) Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin2  + cos2  = 1 tan2 + 1 = sec2 cot2  + 1 = csc2 

  20. Draw a right triangle with an angle  suchthat 4 = sec  = = . 4 hyp θ adj 1 sin  = csc  = = cos  = sec  = = 4 tan  = = cot  = Example: Given 1 Trig Function, Find Other Functions Example: Given sec  = 4, find the values of the other five trigonometric functions of  . Use the Pythagorean Theorem to solve for the third side of the triangle.

  21. Applications Involving Right Triangles The angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression.

  22. Using Trigonometry to Solve a Right Triangle A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument? Figure 4.33

  23. Solution where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78.3  115(4.82882)  555 feet.

  24. How DMS (degrees, minutes and seconds) work 3600

  25. Convert 27 15' 32.4" to decimal degrees

  26. Or you can use a calculator Convert 27 15' 32.4" to decimal degrees • Hit 2nd apps (angle) to find your homepage for DMS • Use Alpha/Plus (Quote) for the seconds symbol Convert 27.259 to DMS

  27. H Dub • 4-3 Page 308 #9-25odd, 29-42all, 43-57odd, 63, 66-68

  28. Find x and y

  29. Find x and y

  30. What if the hypotenuse was 1? • Find x and y

  31. Stuff we went over in 4.2

  32. Some old geometry favorites… Let’s look at the trigonometric functions of a few familiar triangles…

  33. Consider an isosceles right triangle with two sides of length 1. 45 1 45 1 The Pythagorean Theorem implies that the hypotenuse is of length . Geometry of the 45-45-90 Triangle Geometry of the 45-45-90 triangle

  34. Example: Trig Functions for  45 1 sin 45 = = = cos 45 = = = 45 1 tan 45 = = = 1 cot 45 = = = 1 adj opp hyp adj hyp hyp adj sec 45 = = = csc 45 = = = opp adj opp Calculate the trigonometric functions for a 45 angle.

  35. Geometry of the 30-60-90 Triangle 2 2 60○ 60○ 2 Use the Pythagorean Theorem to find the length of the altitude, . Geometry of the 30-60-90 triangle Consider an equilateral triangle with each side of length 2. 30○ 30○ The three sides are equal, so the angles are equal; each is 60. The perpendicular bisector of the base bisects the opposite angle. 1 1

  36. Example: Trig Functions for  30 2 1 30 sin 30 = = cos 30 = = tan 30 = = = cot 30 = = = adj opp hyp adj csc 30 = = = 2 sec 30 = = = hyp hyp adj opp adj opp Calculate the trigonometric functions for a 30 angle.

  37. Example: Using Trigonometric Identities hyp a Side a is opposite θ and also adjacent to 90○– θ . 90○– θ θ b sin  = and cos(90 ) = . So, sin  = cos (90 ). Note sin  = cos(90 ), for 0 <  < 90 Note that  and 90  are complementary angles. Note : These functions of the complements are called cofunctions.

  38. Example: Trig Functions for  60 2 60○ 1 sin 60 = = cos 60 = = cot 60 = = = tan 60 = = = adj opp hyp adj sec 60 = = = 2 csc 60 = = = hyp hyp adj opp adj opp Calculate the trigonometric functions for a 60 angle.

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