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1 Right Triangle Trigonometry Pre-Calculus

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Todays Objective Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions Begin learning some of the Trigonometric identities 2

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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. What You Should Learn

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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).

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5 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. opp adj hyp θ

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

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7 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

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

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

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10 Note : These functions of the complements are called cofunctions. 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. a hyp b θ 90 – θ sin = and cos (90 ) =. So, sin = cos (90 ). Example: Using Trigonometric Identities

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11 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 )

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

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

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14 Quotient Identities

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Pythagorean Identities Three additional identities that we will use are those related to the Pythagorean Theorem: 15 Pythagorean Identities sin 2 + cos 2 = 1 tan = sec 2 cot = csc 2

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16 Some old geometry favorites… Lets look at the trigonometric functions of a few familiar triangles…

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

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18 Calculate the trigonometric functions for a 45 angle. Examp le: Trig Functi ons for csc 45 = = = opp hyp sec 45 = = = adj hyp cos 45 = = = hyp adj sin 45 = = = cot 45 = = = 1 opp adj tan 45 = = = 1 adj opp

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19 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. Geometry of the triangle Use the Pythagorean Theorem to find the length of the altitude,. Geometry of the Triangle

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20 Calculate the trigonometric functions for a 30 angle Example: Trig Functions for 30 csc 30 = = = 2 opp hyp sec 30 = = = adj hyp cos 30 = = hyp adj tan 30 = = = adj opp cot 30 = = = opp adj sin 30 = =

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21 Calculate the trigonometric functions for a 60 angle Example: Trig Functions for 60 csc 60 = = = opp hyp sec 60 = = = 2 adj hyp cos 60 = = hyp adj tan 60 = = = adj opp cot 60 = = = opp adj sin 60 = =

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Some basic trig values SineCosineTangent 30 0 / / /3 22

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IDENTITIES WE HAVE REVIEWED SO FAR… 23

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24 Fundamental Trigonometric Identities 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 ) Reciprocal Identities sin = 1/csc cos = 1/sec tan = 1/cot cot = 1/tan sec = 1/cos csc = 1/sin Quotient Identities tan = sin /cos cot = cos /sin Pythagorean Identities sin 2 + cos 2 = 1 tan = sec 2 cot = csc 2 Fundamental Trigonometric Identities for

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25 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. tan = = cot =sin = csc = = cos = sec = = 4 θ 4 1 Draw a right triangle with an angle such that 4 = sec = =. adj hyp Example: Given 1 Trig Function, Find Other Functions

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Using the calculator Function Keys Reciprocal Key Inverse Keys 26

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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 How tall is the Washington Monument? Figure 4.33

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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.

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

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