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Pre-Calculus Monday, April 20 Right Triangle Trigonometry 1 Today’s Objective Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions Begin learning some of the Trigonometric identities 2 What You Should Learn • Evaluate trigonometric functions of acute angles. • Use fundamental trigonometric identities. • Use a calculator to evaluate trigonometric functions. • Use trigonometric functions to model and solve real-life problems. Plan Questions from last week? Notes! Guided Practice Homework 4 Right Triangle Trigonometry Trigonometry is based upon ratios of the sides of right triangles. The ratio of sides in triangles with the same angles is consistent. The size of the triangle does not matter because the triangles are similar (same shape different size). 5 The six trigonometric functions of a right triangle, with an acute angle θ, are defined by ratios of two sides of the triangle. hyp opp θ The sides of the right triangle are: the side opposite the acute angle the side adjacent to the acute angle adj θ θ, and the hypotenuse of the right triangle. 6 hyp The trigonometric functions are opp θ adj sine, cosine, tangent, cotangent, secant, and cosecant. opp Sinθ = cos θ= adj tan θ= opp hyp hyp adj Csc θ= hyp opp sec θ= hyp adj cot θ= adj opp Note: sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals. 7 Reciprocal Functions Another way to look at it… sin = 1/csc cos = 1/sec tan = 1/cot csc = 1/sin sec = 1/cos cot = 1/tan 8 Given 2 sides of a right triangle you should be able to find the value of all 6 trigonometric functions. Example: 5 12 9 Calculate the trigonometric functions for . Calculate the trigonometric functions for . 5 The six trig ratios are Sin θ = Cos θ = Tanθ = Cot θ = Sec θ = Cscθ = 4 5 3 5 4 3 3 4 5 3 5 4 3 sin α = 5 4 cos α = 5 3 tan α = 4 4 cot α = 3 5 sec α = 4 5 csc α = 3 4 3 What is the relationship of α and θ? They are complementary (α = 90 – θ) 10 Note sin = cos(90 ), for 0 < < 90 Note that and 90 are complementary angles. Side a is opposite θ and also adjacent to 90○– θ . sin = a and cos (90 ) = a . b b So, sin = cos (90 ). hyp 90○– θ a θ b Note : These functions of the complements are called cofunctions. 11 Cofunctions sin = cos (90 ) sin = cos (π/2 ) cos = sin (90 ) cos = sin (π/2 ) tan = cot (90 ) tan = cot (π/2 ) cot = tan (90 ) cot = tan (π/2 ) sec = csc (90 ) csc = sec (90 ) sec = csc (π/2 ) csc = sec (π/2 ) 12 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. 13 Quotient Identities hyp opp θ adj Sin θ = opp hyp cos θ = adj hyp tan θ = opp adj opp sin hyp opp hyp opp tan cos adj hyp adj adj hyp The same argument can be made for cot… since it is the reciprocal function of tan. 14 Quotient Identities sin tan cos cos cot sin 15 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 16 Some old geometry favorites… Let’s look at the trigonometric functions of a few familiar triangles… 17 Geometry of the 45-45-90 triangle Consider an isosceles right triangle with two sides of length 1. 45 2 1 12 12 2 45 1 The Pythagorean Theorem implies that the hypotenuse is of length 2 . Remember a2 + b2 = c2 18 Calculate the trigonometric functions for a 45° angle. 2 1 45 1 opp sin 45° = hyp 1 = 2 opp tan 45° = adj 1 = 1 2 = 2 = 1 2 hyp sec 45° = = =2 1 adj 1 2 adj cos 45° = = = 2 hyp 2 adj 1 cot 45° = = = 1 opp 1 2 hyp csc 45° = = opp 1 =2 19 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°. 2 The perpendicular bisector of the base bisects the opposite angle. 60○ 2 3 1 60○ 2 1 Use the Pythagorean Theorem to find the length of the altitude, 3 . 20 Calculate the trigonometric functions for a 30 angle. 2 1 30 3 opp sin 30° = hyp 1 = 2 3 adj cos 30° = = 2 hyp 1 opp tan 30° = = adj 3 3 = 3 adj cot 30° = opp 3 = 1 =3 2 hyp sec 30° = = 3 adj 2 3 = 3 hyp csc 30° = opp 2 = 1 = 2 21 Calculate the trigonometric functions for a 60 angle. 2 3 60○ 1 sin 60 = opp 3 = hyp 2 cos 60 = tan 60 = 3 opp = = 3 1 adj 3 1 cot 60 = adj = = opp 3 3 hyp 2 sec 60 = = = 2 adj 1 1 adj = 2 hyp 2 2 3 hyp csc 60 = = = opp 3 3 22 Some basic trig values Sine 300 /6 450 /4 600 /3 Cosine Tangent 1 2 3 2 3 3 2 2 2 2 1 3 2 1 2 3 23 IDENTITIES WE HAVE REVIEWED SO FAR… 24 Fundamental Trigonometric Identities Reciprocal Identities sin = 1/csc cot = 1/tan cos = 1/sec sec = 1/cos tan = 1/cot csc = 1/sin Co function Identities sin = cos(90 ) sin = cos (π/2 ) tan = cot(90 ) tan = cot (π/2 ) sec = csc(90 ) sec = csc (π/2 ) Quotient Identities tan = sin /cos cos = sin(90 ) cos = sin (π/2 ) cot = tan(90 ) cot = tan (π/2 ) csc = sec(90 ) csc = sec (π/2 ) cot = cos /sin Pythagorean Identities sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2 25 Example: Given sec = 4, find the values of the other five trigonometric functions of . Draw a right triangle with an angle such 4 4 hyp that 4 = sec = = . adj 1 Use the Pythagorean Theorem to solve for the third side of the triangle. sin = 15 4 cos = 1 4 tan = 15 = 15 1 15 θ 1 1 = 4 sin 15 1 sec = =4 cos 1 cot = 15 csc = 26 Using the calculator Function Keys Reciprocal Key Inverse Keys 27 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 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. 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. Homework 4-2 Practice 1 1-17 ODD 31 Tuesday, April 21, 2015 BENCHMARK TOMORROW 32 Kahoot all day long! Here’s the deal… 1. 2. 3. 4. 5. You need a PENCIL and PAPER You will be GRADED for participating so make sure your Kahoot name is your real name If you get kicked out, you must log back in If you do not have an ipad/smart phone, you may work in teams of TWO or do your work on a piece of paper and turn that in THIS IS REQUIRED. 33 Review Exponents and Log Converting Logs and Exponents https://play.kahoot.it/#/k/68a4661b-b90b4987-820e-956c7e5af6bd Log and Inverses https://play.kahoot.it/#/k/45a02d91-aa36485f-ba52-436768d981f7 34 Review Intro to Trig Right Triangle Trig and Angle Measures https://play.kahoot.it/#/k/8858d7ed-b09246c6-bc5d-b16c044665c0 Radians, Degrees, Arc Length https://play.kahoot.it/#/k/63ac7b20-59ce4fd4-9266-954797b1b39a 35 Wednesday BENCHMARK! 36 Thursday, April 23 37 Do Now – How was the benchmark? What can you improve? Benchmark Data 39 Benchmark Data 1st Period 40 Analysis By Question 41 Analysis By Question 42 Analysis By Question 43 Analysis By Question 44 Analysis By Question 45 Benchmark Data 2nd Period 46 Analysis By Question 47 Analysis By Question 48 Analysis By Question 49 Analysis By Question 50 Analysis By Question 51 Special Angle Names Angle of Elevation From Horizontal Up Angle of Depression From Horizontal Down Angle of Elevation and Depression Imagine you are standing here. The angle of elevation is measured from the horizontal up to the object. Angle of Elevation and Depression The angle of depression is measured from the horizontal down to the object. Constructing a right triangle, we are able to use trig to solve the triangle. Guided Practice! Follow along on your handout! Lighthouse & Sailboat Suppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat? x 5.7o 150 ft. Construct a triangle and label the known parts. Use a variable for the unknown value. 150 ft. Lighthouse & Sailboat Suppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat? x 150 ft. Set up an equation and solve. 5.7o Lighthouse & Sailboat tan(5.7 o ) 150 x x x tan(5.7 ) 150 o x 150 tan(5.7o ) 150 ft. Remember to use degree mode! x is approximately 1,503 ft. 5.7o River Width A surveyor is measuring a river’s width. He uses a tree and a big rock that are on the edge of the river on opposite sides. After turning through an angle of 90° at the big rock, he walks 100 meters away to his tent. He finds the angle from his walking path to the tree on the opposite side to be 25°. What is the width of the river? Draw a diagram to describe this situation. Label the variable(s) River Width We are looking at the “opposite” and the “adjacent” from the given angle, so we will use tangent d tan(25 ) 100 Multiply by 100 on both sides 100 tan(25 ) d d 46.63 meters Subway The DuPont Circle Metrorail Station in Washington DC has an escalator which carries passengers from the underground tunnel to the street above. If the angle of elevation of the escalator is 52° and a passenger rides the escalator for 188 ft, find the vertical distance between the tunnel and the street. In other words, how far below street level is the tunnel? Subway We are looking at the “opposite” and the “hypotenuse” from the given angle so we will use sine h sin(52 ) 188 Multiply by 188 on each side 188sin(52 ) h h 148.15 feet Building Height A spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35o. The angle of elevation to the top of the spire is 38o. How tall is the spire? Construct the required triangles and label. 38o 35o 500 ft. Building Height Write an equation and solve. Total height (t) = building height (b) + spire height (s) Solve for the spire height. s t Total Height t tan(38 ) 500 o b 500 tan(38 ) t o 38o 35o 500 ft. Building Height Write an equation and solve. Building Height b o tan(35 ) 500 s 500 tan(35o ) b t b 38o 35o 500 ft. Building Height Write an equation and solve. Total height (t) = building height (b) + spire height (s) 500 tan(38o ) t 500 tan(35o ) b s 500 tan(38 ) 500 tan(35 ) s 500 tan(38o ) 500 tan(35o ) s o The height of the spire is approximately 41 feet. o t b 38o 35o 500 ft. Mountain Height A hiker measures the angle of elevation to a mountain peak in the distance at 28o. Moving 1,500 ft closer on a level surface, the angle of elevation is measured to be 29o. How much higher is the mountain peak than the hiker? Construct a diagram and label. 1st measurement 28o. 2nd measurement 1,500 ft closer is 29o. Mountain Height Adding labels to the diagram, we need to find h. h ft 28o 1500 ft 29o x ft Write an equation for each triangle. Remember, we can only solve right triangles. The base of the triangle with an angle of 28o is 1500 + x. h tan 28 1500 x o h tan 29 x o Mountain Height Now we have two equations with two variables. Solve by substitution. h tan 28 1500 x o h tan 29 x o Solve each equation for h. (1500 x) tan(28o ) h x tan(29o ) h Substitute. (1500 x) tan(28o ) x tan(29o ) Mountain Height (1500 x) tan(28o ) x tan(29o ) Solve for x. Distribute. 1500 tan(28o ) x tan(28o ) x tan(29o ) Get the x’s on one side and factor out the x. 1500 tan(28o ) x tan(29o ) x tan(28o ) 1500 tan(28o ) x tan(29o ) tan(28o ) Divide. 1500 tan(28o ) x o o tan(29 ) tan(28 ) Mountain Height However, we were to find the height of the mountain. Use one of the equations solved for “h” to solve for the height. x tan(29o ) h 1500 tan 28o tan 29o tan 29 tan 28 o o 19,562 The height of the mountain above the hiker is 19,562 ft.