4.1  Trigonometric Functions of Acute Angles


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1 4.1  Tigonometic Functions of cute ngles a is a halfline that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one a as the initial a and the othe a as the teminal a, the measue of the angle is the amount of otation needed to make the initial a coincide with the teminal a. common unit of the measue of an angle is a degee. We define one degee (1 ) to be 1/60 of a complete counteclockwise otation. Thee ae 60 in a complete counteclockwise otation. It is common to use the Geek lette (theta) to epesent an angle measue. n acute angle measues between 0 and 90, and an obtuse angle measues between 90 and180. ight tiangle is a tiangle that contains a90 angle. pg. 179 Definition of the Tigonometic Functions If θ (theta) is an acute angle in a ight tiangle, then Name of Function bbeviation Ratio sine of angle θ sin θ opposite hpotenuse cosecant of angle θ csc θ hpotenuse opposite cosine of angle θ secant of angle θ tangent of angle θ cotangent of angle θ cos θ sec θ tan θ cot θ adjacent hpotenuse hpotenuse adjacent opposite adjacent adjacent opposite Pogess Check Find the values of the emaining tigonometic functions of the acute angle θ if cos. 5 Pogess Check In tiangle BC with C 90, if 8 sin and a 40 find c. 17 1
2 pg. 00 Eact Tigonometic Values fo 0, 45, and 60 θ Sin θ Csc θ Cos θ Sec θ Tan θ Cot θ tiangle tiangle Pogess Check 4 Find the eact values of sin 60, cot 0, csc0, and cos 45 pg. 01 Cofunction Popeties Fo an acute angle θ, sin(90 θ) = cos θ tan(90 θ) = cot θ sec(90 θ) = csc θ cos(90 θ) = sin θ cot(90 θ) = tan θ csc(90 θ) = sec θ Pogess Check 5 Epess cos4 as a function of the acute angle complimenta to 4. Pogess Check 6 Evaluate each epession b calculato to fou decimal places. a. tan 6 b. sec16.7 c. sin81 50 Pogess Check 8 Use a calculato to appoimate the acute angle that satisfies the given equation. Wite solutions to the neaest tenth of a degee and to the neaest 10 minutes. a. cos 0.78 b. csc 1.448
3 4.  Right Tiangle pplications To solve ight tiangle means to find the measues of the two acute angles and the length of the thee sides of the tiangle. To accomplish this, at least two of these five values must be known, and one o moe must be a side length. How to Round when Solving a Right Tiangle ccuac of Sides Two significant digits Thee significant digits Fou significant digits ccuac of ngles Neaest degee Neaest 10 minutes o tenth of a degee Neaest minute o hundedth of a degee Diagam of an BC Right Tiangle Solving a Right Tiangle: ngleside Case Pogess Check Solve the ight tiangle BC in which 9 40' and c 7.5 ft.
4 Solving a Right Tiangle: Two Sides Case Pogess Check Solve the ight tiangle BC in which b 1.0 ft. and c 19.0 ft. Pogess Check 4 ladde leans against the side of a building and makes an angle of 7.0 with the gound. If the ladde is 5.0 ft. long, then find the height the ladde eaches on the building. Pogess Check 5 suveo stands on a cliff 175 ft. above a ive. If the angle of depession to the ive s edge on the opposite bank is8.4, how wide is the ive at this point? 4
5 4.  Tigonometic Functions of Geneal ngles pg. 14 Definition of the Tigonometic Functions If θ (theta) is an angle in standad position, and if (, ) is an point on the teminal a of θ [ecept (0,0)], then Name of Function bbeviation Ratio sine of angle θ sin θ cosecant of angle θ csc θ ( 0) cosine of angle θ cos θ secant of angle θ sec θ ( 0) tangent of angle θ tan θ cotangent of angle θ cot θ ( 0) Pogess Check 1 Find the values of the si tigonometic functions of angle if, 1 is a point on the teminal a of. 5
6 Pogess Check Find the values of the emaining tigonometic functions if sin and tan 0. 4 Quadtantal ngles ae angles that have a teminal a that coincides with one of the aes. n quadantal angle can be epessed as the poduct of 90 o and some intege. In geneal, if two angles have the same teminal a, the ae called coteminal, and the tigonometic functions of coteminal angles ae equal. pg. 17 θ Sin θ Csc θ Cos θ Sec θ Tan θ Cot θ 0 0 Undefined Undefined Undefined Undefined Undefined Undefined Undefined Undefined 0 Refeence angles ae useful in evaluating tigonometic functions fo angles that ae not quadantal angles. The efeence angle fo an angle θ is defined to be the positive acute angle fomed b the teminal a of θ and the hoizontal ais. To Evaluate Tigonometic Functions Fo nonquadantal angles: 1. Find the efeence angle fo the given angle.. Find the tigonometic value of the efeence angle using the appopiate function. If the efeence angle is 0 o, 45 o, o 60 o, the eact answe is pefeable.. Detemine the coect sign accoding to the teminal a of the angle. 6
7 Pogess Check Find the si tigonometic functions of 180, using the definition of the tigonometic functions. Pogess Check 4 Find the eact value of each epession. a. cos70 b. cot 90 Pogess Check 5 Find the eact value of sin00. Pogess Check 6 Find the eact value of sin15. 7
8 4.4  Intoduction to Tigonometic Equations pg. 5 Quadant Solution 1 efeence angle 180 o efeence angle 180 o + efeence angle 4 60 o efeence angle Pogess Check Find the eact values of 0 60 tue statement. fo which the equation cos 1 0 is a 8
9 Pogess Check To the neaest 10 minutes, appoimate the values of 0 60 equation sin is a tue statement. fo which the Pogess Check 4 Solve tan 5 fo Round off answes to the neaest tenth of a degee. Pogess Check 5 ppoimate all the solutions to sec 7 1, to the neaest 10 minutes. 9
10 Chapte 4 Oveall Review atios sin cos tan O H H O csc = sec cot H O H O a b c USE WHEN: 1) Given diagam of a ight tiangle with side lengths find ) Given one tig atio find emaining tig atios *If not pefect squae oot leave H O sides find c B SOLVING RIGHT TRINGLE BC : 1) Given one angle and one side find one angle and two a ) Given two sides find one side and two angles ) Given one tig atio and one side of simila tiangle anothe side of the simila tiangle *If sides ae given with significant digits, angles must be to the neaest degee. C *If sides ae given with significant digits, b angles must be to the neaest tenth of a degee o neaest ten minutes 10
11 sin cos tan csc = sec cot o USE WHEN: 1) Given a point on the teminal a ) Given an equation and quadant ) Given one tig atio and quadant NOTE: and ma be negative depending what quadant the teminal a is in, but is neve negative. *If not pefect squae oot leave S T C b Finding the values of 0 60g fo which a tigonometic equation is a tue statement: 1) Solve fo the tigonometic function (if necessa). ) Detemine which two quadants contain the teminal a of. ) Detemine the efeence angle: * Use tig invese buttons on calculato to find the efeence angle but don t use negative sign if tig atio has one. 4) Detemine the two appopiate values of b dawing o use the following, Q1: Ref Q : 180 Ref Q: 180 Ref Q : 60 Ref 4 Chapte 9 Law of Cosines a b c bc Cos Law of Sines sin sin B sinc a b c 11
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