Right Triangle Trigonometry 6.2

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

3 The Six Trigonometric Functions

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

10 Example – Solution cont’d 5

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

12 Consider an isosceles right triangle with two sides of length The Pythagorean Theorem implies that the hypotenuse is of length. Geometry of the Triangle

13 Calculate the trigonometric functions for a 45  angle 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 Triangle

14 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  ○ Use the Pythagorean Theorem to find the length of the altitude,. Geometry of the Triangle

15 Calculate the trigonometric functions for a 30  angle 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 Triangle

16 Calculate the trigonometric functions for a 60  angle ○ 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 Triangle

17 The Six Trigonometric Functions 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.

18 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 

19 Trigonometric Identities

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

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

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 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 = Simplify.

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 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.

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

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

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.

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

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 Finding Trigonometric Function Values Using a Calculator

33 Function Values Using a Calculator Calculators are capable of finding trigonometric function values. When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode. Remember that most calculator values of trigonometric functions are approximations.

34 Example: Finding Function Values with a Calculator a) Convert 38  to decimal degrees. b) cot  Use the identity cot 

35 Angle Measures Using a Calculator Calculators have three inverse functions. If x is an appropriate number, then gives the measure of an angle whose sine, cosine, or tangent is x.

36 Example: Using Inverse Trigonometric Functions to Find Angles Use a calculator to find an angle in the interval that satisfies each condition. Using the degree mode and the inverse sine function, we find that an angle having sine value is We write the result as

37 Example: Using Inverse Trigonometric Functions to Find Angles continued Use the identity Find the reciprocal of to get Now find using the inverse cosine function. The result is

38 Applications Involving Right Triangles

39 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.

40 Solving an Applied Trigonometry Problem Step 1Draw a sketch, and label it with the given information. Label the quantity to be with a variable. Step 2Use the sketch to write an equation relating the given quantities to the variable. Step 3Solve the equation, and check that your answer makes sense.

41 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.

42 Angle of Elevation and Depression

43 Example: Application The length of the shadow of a tree m tall is m. Find the angle of elevation of the sun. Draw a sketch. The angle of elevation of the sun is  m m B

44 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?

45 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( )  555 feet.

46 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:

47 Your Turn: a = yards

48 Your Turn: AOE = 48 ̊

49 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

50 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?

51 Solution: 20 = 9.46 ̊

52 Example: Solving a Problem Involving Angles of Elevation Sean wants to know the height of a Ferris wheel. From a given point on the ground, he finds the angle of elevation to the top of the Ferris wheel is He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is Find the height of the Ferris wheel.

53 Example: Solving a Problem Involving Angles of Elevation continued The figure shows two unknowns: x and h. Since nothing is given about the length of the hypotenuse, of either triangle, use a ratio does not involve the hypotenuse, tangent. In triangle ABC, In triangle BCD, x C B h DA 75 ft

54 Example: Solving a Problem Involving Angles of Elevation continued Since each expression equals h, the expressions must be equal to each other. Solve for x. Distributive Property Factor out x. Get x-terms on one side. Divide by the coefficient of x.

55 We saw above that Substituting for x. tan 42.3 = and tan 25.4 = So, tan tan 25.4 = = and The height of the Ferris wheel is approximately 74 ft. Example: Solving a Problem Involving Angles of Elevation continued

56 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.

57 Solution: 3.5 ̊ 9 ̊ h x13 h = 1.3 miles

58 Assignment Sec. 6.2, Pg. 467 – 470: #1 – 65 odd