3 October Kidoguchi, Kenneth

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1 5. Tigonometic Functions: Unit Cicle Appoach Objectives. Find the Eact Values of the Tigonometic Functions Using a Point on the Unit Cicle. Find the Eact Values of the Tigonometic Functions of Quadantal Angles 3. Find the Eact Values of the Tigonometic Functions of p/ = 5º. Find the Eact Values of the Tigonometic Functions of p/6 = 30º and p/3 = 60º 5. Find the Eact Values of the Tigonometic Functions fo intege multiples of p/6 = 30º, p/ = 5º, and p/3 = 60º 6. Use a Calculato to Appoimate the Values of a Tigonometic Function 7. Use a Cicle of Radius to Evaluate the Tigonometic Functions 3 Octobe 07 Kidoguchi, Kenneth

2 5. Tigonometic Functions: Unit Cicle Appoach The Length of an Ac of a Cicle s 3 3 s 3 s (, 0) Angle in adians: ac length s adius p evolution p N.B.: An angle in adians is length ove length, hence a dimensionless quantit. Fo a cicle of adius, a cental angle in adians subtends an ac of length s such that: s = 3 Octobe 07 Kidoguchi, Kenneth

3 5. Tigonometic Functions: Unit Cicle Appoach Unit Cicle and a Point on a Unit Cicle A unit cicle is a cicle of adius unit centeed at the oigin of a Catesian (i.e., ectangula) coodinate sstem. Let t be a eal numbe and let P = (,) be the point on the unit cicle that coesponds to t. P = (, ) s = t > 0 s = t < 0 (, 0) (, 0) P = (, ) 3 Octobe 07 3 Kidoguchi, Kenneth

4 5. Tigonometic Functions: Unit Cicle Appoach Tigonometic Functions on a Unit Cicle Let t be a eal numbe and let P = (,) be the point on the unit cicle (i.e., a cicle with = ) that coesponds to t. The sine function associates with t the sin( t) -coodinate of P and is denoted b: The cosine function associates with t the cos( t) -coodinate of P and is denoted b: If 0, the tangent function associates with t tan( t) the atio of the -coodinate to the - coodinate of P and is denoted b: If 0, the cosecant function defined as: csc( t) If 0, the secant function defined as: sec( t) If 0, the cotangent function defined as: cot( t) 3 Octobe 07 Kidoguchi, Kenneth

5 5. Tigonometic Functions: Unit Cicle Appoach. Eact Values of the Tig Functions Using a Point on the Unit Cicle Let t be a eal numbe and let cicle that coesponds to t. P 6 =, be a point on the unit 5 5 Find the values of sin(t), cos(t), tan(t), csc(t), sec(t), and cot(t). 3 Octobe 07 5 Kidoguchi, Kenneth

6 5. Tigonometic Functions: Unit Cicle Appoach Tigonometic Functions of Angles = t (, ) Quad II 3 (, ) Quad I Quad IV = t (, 0) sin cos 3 3 sin 0 cos 0 Quad III (, ) 3 = t 3 = t ( 3, 3 ) 3 Octobe 07 7 Kidoguchi, Kenneth

7 5. Tigonometic Functions: Unit Cicle Appoach Tigonometic Functions of Angles - Definition If = t adians, the si tigonometic functions of the angle ae defined as: t t t t t t sin sin, cos cos, tan tan csc csc, sec sec, cot cot 3 Octobe 07 8 Kidoguchi, Kenneth

8 5. Tigonometic Functions: Unit Cicle Appoach. Eact Values of Tigonometic Functions of Quadantal Angles Find eact values of the si tigonometic functions of: a) 0 0º b) p/ 90º 3 Octobe 07 9 Kidoguchi, Kenneth

9 5. Tigonometic Functions: Unit Cicle Appoach. Eact Values of Tigonometic Functions of Quadantal Angles Find eact values of the si tigonometic functions of: a) p 80º b) 3p/ 70º 3 Octobe 07 0 Kidoguchi, Kenneth

10 5. Tigonometic Functions: Unit Cicle Appoach Eact Values of Tigonometic Functions of Quadantal Angles Quadantal Angles (ad) (º ) sin() cos() tan() csc() sec() cot() und und p/ 90 0 und und 0 p und - und 3p/ 70 0 und und 0 und = undefined 3 Octobe 07 Kidoguchi, Kenneth

11 5. Tigonometic Functions: Unit Cicle Appoach Eact Values of Tigonometic Functions of Quadantal Angles Find the eact value of: (a) sin(5p) (b) cos( 50 ) 3 Octobe 07 Kidoguchi, Kenneth

12 5. Tigonometic Functions: Unit Cicle Appoach 3. Eact Values of Tigonometic Functions of = p/ /8 Teminal Side 3/8 (,) /8 p/ 0 (, 0) Initial Side 5/8 7/8 6/8 3 Octobe 07 3 Kidoguchi, Kenneth

13 5. Tigonometic Functions: Unit Cicle Appoach. Eact Values of Tigonometic Functions of = p/3 & p/6 / 3/ / (,) / / 5/ / 3 6/ 7/ 8/ 9/ p/3 / / 0/ (, 0) 0 / 3 p 3 sin 3 p cos 3 3 Octobe 07 5 Kidoguchi, Kenneth

14 5. Tigonometic Functions: Unit Cicle Appoach. Eact Values of Tigonometic Functions of = p/3 & p/6 6/ 5/ / 3/ p/6 / (,) / 0 (, 0) / / 3 3 7/ / p p sin cos 6 3 8/ 9/ 0/ p p 3 cos sin Octobe 07 6 Kidoguchi, Kenneth

15 5. Tigonometic Functions: Unit Cicle Appoach. Eact Values of a Tigonometic Epession Find the eact values of each epession. (a) z sin 80º cos 5º p 3p (b) z sin cosp sin p p (c) z3 csc tan 3 Octobe 07 7 Kidoguchi, Kenneth

16 5. Tigonometic Functions: Unit Cicle Appoach Eact Values of Tigonometic Functions Quadantal Angles (ad) (º ) sin() cos() tan() csc() sec() cot() p/6 30 / 3 / / 3 / 3 3 p/ 5 / / p/ / / 3 / 3 / 3 3 Octobe 07 8 Kidoguchi, Kenneth

17 5. Tigonometic Functions: Unit Cicle Appoach Eample Application Pojectile Motion The path of a pojectile with initial velocit v 0 and tajecto angle is a paabola. The hoizontal distance tavelled b the pojectile is its ange R: R v 0 sin g Whee g is the acceleation due to gavit. The maimum height H of the pojectile is: H v sin 0 g Find R and H if v 0 =50 m/s, = 30, with g = 9.8 m/s. v 0 H R 3 Octobe 07 9 Kidoguchi, Kenneth

18 5. Tigonometic Functions: Unit Cicle Appoach Eample Application Constucting a Rain Gutte A ain gutte is to constucted of aluminum sheets inches wide. Afte making off a length of inches fom each edge, this length is bent up at angle as shown in the figue. The aea A of the opening ma be epessed as a function as: A() = 6 sin() (cos() + ) Find the aea A of the opening fo = 30º, = 5º, and = 60º 3 Octobe 07 0 Kidoguchi, Kenneth

19 5. Tigonometic Functions: Unit Cicle Appoach Eample Application Constucting a Rain Gutte A() = 6 sin() (cos() + ) (º) A() (in ) Appo A() (in ) Octobe 07 Kidoguchi, Kenneth

20 5. Tigonometic Functions: Unit Cicle Appoach 5. Eact Values of Tig Functions fo Intege Multiples of p/6, p/ & p/3 /8 3/8 /8 (,) p/ Teminal Side 0 (, 0) Initial Side / 5/8 6/8 7/8 3 Octobe 07 Kidoguchi, Kenneth sin p / cos p /

21 5. Tigonometic Functions: Unit Cicle Appoach 5. Eact Values of Tig Functions fo Intege Multiples of p/6, p/ & p/3 3/8 (,) /8 (,) sin p / cos p / /8 3 p/ p/ 0 (, 0) 5/8 6/8 7/8 3 Octobe 07 3 Kidoguchi, Kenneth sin p / cos p /

22 5. Tigonometic Functions: Unit Cicle Appoach 5. Eact Values of Tig Functions fo Intege Multiples of p/6, p/ & p/3 3/8 (,) /8 (,) sin p / cos p / sin 3p / /8 3 p/ p/ 0 (, 0) cos 3p / 5/8 7/8 6/8 3 Octobe 07 Kidoguchi, Kenneth

23 5. Tigonometic Functions: Unit Cicle Appoach 5. Eact Values of Tig Functions fo Intege Multiples of p/6, p/ & p/3 3/8 (,) /8 (,) sin cos p / p / sin 3p / /8 5 p/ p/ 0 (, 0) cos 3p / sin 5p / cos 5p / 5/8 7/8 6/8 3 Octobe 07 5 Kidoguchi, Kenneth

24 3 Octobe 07 6 Kidoguchi, Kenneth 5. Tigonometic Functions: Unit Cicle Appoach 5. Eact Values of Tig Functions fo Intege Multiples of p/6, p/ & p/3 /8 3/8 5/8 6/8 7/8 (, 0) 0 (,) /8 (,) / cos / sin p p / 5 cos / 5 sin p p / cos / sin p p / 3 cos / 3 sin p p p/ p/

25 5. Tigonometic Functions: Unit Cicle Appoach 5. Eact Values of Tig Functions fo Intege Multiples of p/ = 5º Find the eact values of: (a) cos 35º (b) tan p (c) sin 35º 3 Octobe 07 7 Kidoguchi, Kenneth

26 5. Tigonometic Functions: Unit Cicle Appoach 5. Eact Values of Tig Functions fo Intege Multiples of p/6 o p/3 Find the eact values of: 7p (a) tan 6 (b) sin 0º (c) cos p 3 (d) sin p sin 3 Octobe 07 8 Kidoguchi, Kenneth

27 5. Tigonometic Functions: Unit Cicle Appoach 6. Calculato to Evaluate a Tigonometic Function Use a calculato to appoimate the value of the tigonometic function ounded to two decimal places: (a) cos 8º p (b) tan (c) sec 0 3 Octobe 07 9 Kidoguchi, Kenneth

28 5. Tigonometic Functions: Unit Cicle Appoach 7. Use a Cicle of Radius to Evaluate a Tigonometic Function Theoem Fo an angle in standad position, let P = (,) be the point on the teminal side of that is also on the cicle + =. Then sin cos tan, 0 csc, 0 sec, 0 cot, 0 3 Octobe Kidoguchi, Kenneth

29 5. Tigonometic Functions: Unit Cicle Appoach Use a Cicle of Radius to Evaluate a Tigonometic Function Find the eact values of each of the si tigonometic functions of an angle if (-3, 3) is a point on its teminal side. 3 Octobe 07 3 Kidoguchi, Kenneth

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