2Today’s ObjectiveReview right triangle trigonometry from Geometry and expand it to all the trigonometric functions Begin learning some of the Trigonometric identities
3What You Should LearnEvaluate 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.
4Right 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).
5The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle.θhypoppThe 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.
6Trigonometric Functions θhypThe trigonometric functions areoppadjsine, cosine, tangent, cotangent, secant, and cosecant.sin = cos = tan =csc = sec = cot =opphypadjNote: sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals.Trigonometric Functions
7Reciprocal 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
8Given 2 sides of a right triangle you should be able to find the value of all 6 trigonometric functions. Example:512
9Example: Six Trig Ratios Calculate the trigonometric functions for .Calculate the trigonometric functions for .435The six trig ratios aresin =sin α =cos =cos α =tan =What is the relationship ofα and θ?tan α =cot =cot α =sec =They are complementary (α = 90 – θ)sec α =csc =csc α =Example: Six Trig Ratios
10Example: Using Trigonometric Identities Note sin = cos(90 ), for 0 < < 90Note that and 90 are complementary angles.Side a is opposite θ and also adjacent to 90○– θ .ahypbθ90○– θsin = and cos (90 ) = .So, sin = cos (90 ).Note : These functions of the complements are called cofunctions.Example: Using Trigonometric Identities
11Cofunctions sin = cos (90 ) cos = sin (90 ) 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 )
12Example: 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.Example: Using Trigonometric Identities
13Quotient Identities hyp opp θ adj adj sin = cos = tan = opp adj The same argument can be made for cot… since it is the reciprocal function of tan.
15Pythagorean Identities Three additional identities that we will use are those related to the Pythagorean Theorem:Pythagorean Identitiessin2 + cos2 = 1tan2 + 1 = sec2 cot2 + 1 = csc2
16Some old geometry favorites… Let’s look at the trigonometric functions of a few familiar triangles…
17Geometry of the 45-45-90 Triangle Consider an isosceles right triangle with two sides of length 1.145The Pythagorean Theorem implies that the hypotenuse is of lengthGeometry of the Triangle
18Example: Trig Functions for 45 Calculate the trigonometric functions for a 45 angle.145sin 45 = = =cos 45 = = =hypadjtan 45 = = = 1adjoppcot 45 = = = 1oppadjsec 45 = = =adjhypcsc 45 = = =opphypExample: Trig Functions for 45
19Geometry of the 30-60-90 Triangle 2Consider an equilateral triangle with each side of length 2.60○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.11Use the Pythagorean Theorem to find the length of the altitude,Geometry of the Triangle
20Example: Trig Functions for 30 Calculate the trigonometric functions for a 30 angle.1230cos 30 = =hypadjsin 30 = =tan 30 = = =adjoppcot 30 = = =oppadjsec 30 = = =adjhypcsc 30 = = = 2opphypExample: Trig Functions for 30
21Example: Trig Functions for 60 Calculate the trigonometric functions for a 60 angle.1260○sin 60 = =cos 60 = =hypadjtan 60 = = =adjoppcot 60 = = =oppadjsec 60 = = = 2adjhypcsc 60 = = =opphypExample: Trig Functions for 60
25Example: Given 1 Trig Function, Find Other Functions Example: Given sec = 4, find the values of the other five trigonometric functions of .Draw a right triangle with an angle such that 4 = sec = = .adjhypθ41Use the Pythagorean Theorem to solve for the third side of the triangle.sin = csc = =cos = sec = = 4tan = = cot =Example: Given 1 Trig Function, Find Other Functions
26Using the calculatorFunction Keys Reciprocal Key Inverse Keys
27Using 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
28Applications 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.
29Solutionwhere x = 115 and y is the height of the monument. So, the height of the Washington Monument isy = x tan 78.3 115( ) 555 feet.