1. Right Triangle Trigonometry 4.3

2. 2  Evaluate trigonometric functions of acute angles, and use a calculator to evaluate trigonometric functions.  Use the fundamental trigonometric identities.  Use trigonometric functions to model and solve real-life problems. Objectives

3. 3 The Six Trigonometric Functions

4. 4 This section introduces the trigonometric functions from a right triangle perspective. Consider a right triangle with one acute angle labeled , as shown below. Relative to the angle , the three sides of the triangle are the hypotenuse, the opposite side (the side opposite the angle  ), and the adjacent side (the side adjacent to the angle  ).

5. 5 The Six Trigonometric Functions Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle . Function ⟹ Reciprocal sine ⟹ cosecant cosine ⟹ secant tangent ⟹ cotangent

6. 6 The Six Trigonometric Functions In the following definitions, it is important to see that 0  <  < 90  (  lies in the first quadrant) and that for such angles the value of each trigonometric function is positive.

7. 7 The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are:  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle. The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. opp adj hyp θ sin  = cos  = tan  = csc  = sec  = cot  = opp hyp adj hyp adj opp adj The Six Trigonometric Functions

8. 8 Example – Evaluating Trigonometric Functions Use the triangle in Figure 1.20 to find the values of the six trigonometric functions of . Solution: By the Pythagorean Theorem, (hyp) 2 = (opp) 2 + (adj) 2 it follows that Figure 1.20

9. 9 Example – Solution So, the six trigonometric functions of  are cont’d 5

10. 10 Example – Solution cont’d 5

11. 11 Calculate the trigonometric functions for . The six trig ratios are 6 8 10  sin  = tan  = sec  = cos  = cot  = csc  = Your Turn:

12. 12 The Six Trigonometric Functions In the Example, you were given the lengths of two sides of the right triangle, but not the angle . Often, you will be asked to find the trigonometric functions of a given acute angle . To do this, construct a right triangle having  as one of its angles.

13. 13 The Six Trigonometric Functions In the box, note that sin 30  = = cos 60 . This occurs because 30  and 60  are complementary angles. In general, it can be shown from the right triangle definitions that cofunctions of complementary angles are equal. That is, if  is an acute angle, then the following relationships are true. sin(90  –  ) = cos  cos(90  –  ) = sin  tan(90  –  ) = cot  cot(90  –  ) = tan  sec(90  –  ) = csc  csc(90  –  ) = sec 

14. 14 Consider an isosceles right triangle with two sides of length 1. 1 1 45 The Pythagorean Theorem implies that the hypotenuse is of length. Geometry of the 45-45-90 Triangle

15. 15 Calculate the trigonometric functions for a 45  angle. 1 1 45 csc 45  = = = opp hyp sec 45  = = = adj hyp cos 45  = = = hyp adj sin 45  = = = cot 45  = = = 1 opp adj tan 45  = = = 1 adj opp Geometry of the 45-45-90 Triangle

16. 16 60 ○ Consider an equilateral triangle with each side of length 2. The perpendicular bisector of the base bisects the opposite angle. The three sides are equal, so the angles are equal; each is 60 . 22 2 11 30 ○ Use the Pythagorean Theorem to find the length of the altitude,. Geometry of the 30-60-90 Triangle

17. 17 Calculate the trigonometric functions for a 30  angle. 1 2 30 csc 30  = = = 2 opp hyp sec 30  = = = adj hyp cos 30  = = hyp adj tan 30  = = = adj opp cot 30  = = = opp adj sin 30  = = Geometry of the 30-60-90 Triangle

18. 18 Calculate the trigonometric functions for a 60  angle. 1 2 60 ○ csc 60  = = = opp hyp sec 60  = = = 2 adj hyp cos 60  = = hyp adj tan 60  = = = adj opp cot 60  = = = opp adj sin 60  = = Geometry of the 30-60-90 Triangle

19. 19 Trigonometric Identities

20. 20 Remember an identity is an equation that is true for all defined values of a variable. We are going to use the identities that we have already established to "prove" or establish other identities. Let's summarize the basic identities we have. Trigonometric Identities

21. 21 Trigonometric Identities In trigonometry, a great deal of time is spent studying relationships between trigonometric functions (identities).

22. 22 Trigonometric Identities Note that sin 2  represents (sin  ) 2, cos 2  represents (cos  ) 2, and so on.

23. 23 Example – Applying Trigonometric Identities Let  be an acute angle such that sin  = 0.6. Find the values of (a) cos  and (b) tan  using trigonometric identities. Solution: a. To find the value of cos , use the Pythagorean identity sin 2  + cos 2  = 1. So, you have (0.6) 2 + cos 2  = 1 cos 2  = 1 – (0.6) 2 cos 2  = 0.64 Substitute 0.6 for sin . Subtract (0.6) 2 from each side. Simplify.

24. 24 Example 5 – Solution cos  = cos  = 0.8. b. Now, knowing the sine and cosine of , you can find the tangent of  to be Use the definitions of cos  and tan  and a right triangle to check these results. Extract positive square root. cont’d = 0.75. Simplify.

25. 25 Your Turn: Let  be an acute angle such that sec  = 3. Find the values of (a) tan  and (b) sin  using trigonometric identities.

26. 26 Verifying or Proving Identities 1.Learn the fundamental identities. 2.Try to rewrite the more complicated side of the equation so that it is identical to the simpler side. 3.It is often helpful to express all functions in terms of sine and cosine and then simplify the result. 4.Usually, any factoring or indicated algebraic operations should be performed. For example, 5.As you select substitutions, keep in mind the side you are not changing, because it represents your goal. 6.If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x.

27. Prove the following identity: In proving an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match. Let's sub in here using reciprocal identity We often use the Pythagorean Identities solved for either sin 2  or cos 2 . sin 2  + cos 2  = 1 solved for sin 2  is sin 2  = 1 - cos 2  which is our left-hand side so we can substitute. We are done! We've shown the LHS equals the RHS

28. Prove the following identity: Let's sub in here using reciprocal identity and quotient identity Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom We worked on LHS and then RHS but never moved things across the = sign combine fractions FOIL denominator

29. Your Turn: Verifying an Identity ExampleVerify that the following equation is an identity. cot x + 1 = csc x(cos x + sin x) Analytic Solution Since the side on the right is more complicated, we work with it. Original identity Distributive property The given equation is an identity because the left side equals the right side.

30. Your Turn: Verifying an Identity ExampleVerify that the following equation is an identity. Solution

31. 31 Hints for Establishing Identities Get common denominators. If you have squared functions look for Pythagorean Identities. Work on the more complex side first. If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity. When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities. Have fun with these---it's like a puzzle, can you use identities and algebra to get them to match!

32. 32 Applications Involving Right Triangles

33. 33 If the angle is not exactly to the next degree it can be expressed as a decimal (most common in math) or in degrees, minutes and seconds (common in surveying and some navigation). 1 degree = 60 minutes 1 minute = 60 seconds  = 25°48'30" degrees minutes seconds To convert to decimal form use conversion fractions. These are fractions where the numerator = denominator but two different units. Put unit on top you want to convert to and put unit on bottom you want to get rid of. Let's convert the seconds to minutes 30"= 0.5' Degrees, Minutes and Seconds

34. 1 degree = 60 minutes1 minute = 60 seconds  = 25°48'30" Now let's use another conversion fraction to get rid of minutes. 48.5'=.808° = 25°48.5'= 25.808°

35. 35 Your Turn: Convert to DMS or Decimal 5°40’ 12” 16.35° 16°18’ 73.56 ̊ 5.67 ̊ 16 ̊ 21’ 16.3 ̊ 73 ̊ 33’ 36”

36. 36 Applications Involving Right Triangles Many applications of trigonometry involve a process called solving right triangles. In this type of application, you are usually given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or you are given two sides and are asked to find one of the acute angles.

37. 37 Applications Involving Right Triangles In some applications, the angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. In other applications you may be given the angle of depression, which represents the angle from the horizontal downward to an object.

38. 38 Example – Using Trigonometry to Solve a Right Triangle A surveyor is standing 115 feet from the base of the Washington Monument, as shown in figure below. The surveyor measures the angle of elevation to the top of the monument as 78.3 . How tall is the Washington Monument?

39. 39 Example – Solution From the figure, you can see that 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.

40. 40 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? 200 x 60° cos 60° x (cos 60°) = 200 x X = 400 yards Example:

41. 41 Your Turn: A six-foot person standing 20 feet from a streetlight casts a 10-foot shadow. What is the height of the streetlight? Height = 18 ft

42. 42 Your Turn: A ramp 20 feet in length rises to a loading platform that is 3 ⅓ feet off the ground. –Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the angle of elevation of the ramp. –Use a trigonometric function to write an equation involving the unknown quantity. –What is the angle of elevation of the ramp?

43. 43 Solution: 20 = 9.46 ̊

44. 44 Your Turn: In traveling across flat land, you notice a mountain directly in from of you. Its angle of elevation (to the peak) is 3.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9°. Approximate the height of the mountain.

45. 45 Solution: 3.5 ̊ 9 ̊ h x13 h = 1.3 miles

46. 46 Assignment Pg. 284 – 287: #1 – 55 odd, 63 – 81 odd