5.3 Properties of Trigonometric Functions Objectives

Transcription

1 Objectives. Determine the Domain and Range of the Trigonometric Functions. 2. Determine the Period of the Trigonometric Functions. 3. Determine the Signs of the Trigonometric Functions in a Given Quadrant. 4. Find the Values of the Trigonometric Functions Using Fundamental Identities. 5. Find the Eact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle 6. Use Even-Odd Properties to Find Eact Values of the Trigonometric Functions. 3 April 208 Kidoguchi, Kenneth

2 . The Domain and Range of the Trigonometric Functions For a unit circle, i.e. a circle of radius r = : sin csc, 0 cos sec, 0 tan, 0 cot, 0 3 April Kidoguchi, Kenneth

3 . The Domain and Range of the Trigonometric Functions Function Smbol Domain Range sine f() = sin() θ R - f() cosine f() = cos() θ R - f() tangent f() = tan() θ R ^ θ n π 2 - < f() < cosecant f() = csc() θ R ^ θ nπ f() < - f() > secant f() = sec() θ R ^ θ n π 2 f() < - f() > cotangent f() = cot() θ R ^ θ nπ - < f() < n 3 April Kidoguchi, Kenneth

4 Co-terminal Angles 5/2 4/2 3/2 2/2 /2 Let, and are said to be 2 co-terminal because n2 n 2 0,, 2, 3 6/2 0 7/2 /2 8/2 9/2 0/2 3 April Kidoguchi, Kenneth

5 2. The Period of the Trigonometric Functions 3 P, cos sin 3 2 (, 0) 2 3 cos sin April Kidoguchi, Kenneth

6 2. The Period of the Trigonometric Functions sin 2k sin and cos 2k cos, k A function f is called periodic if there is a positive number p such that, whenever is the domain of f, so is + p, and f( + p) = f() If there is a smallest such number p, this smallest value is called the (fundamental) period of f. Periodic Properties n sin( + n 2 ) = sin() cos( + n 2 ) = cos() tan( + n ) = tan() csc( + n 2 ) = csc() sec( + n 2 ) = sec() cot( + n ) = cot() 3 April Kidoguchi, Kenneth

7 Finding Eact Values Periodic Properties - Eample Find the eact value of: (a) sin 390º (b) 9 tan 4 (c) cos 7 3 April Kidoguchi, Kenneth

8 4. Values of the Trig Functions Using Fundamental Identities If P = (,) is the point on a circle of radius r corresponding to, then For an angle in standard position, let P = (,) be the point on the terminal side of that is also on the circle = r 2. Then sin cos tan, 0 r r r r csc, 0 sec, 0 cot, 0 If P = (,) is the point on the unit circle corresponding to, then For an angle in standard position, let P = (,) be the point on the terminal side of that is also on the circle =. Then sin cos tan, 0 csc, 0 sec, 0 cot, 0 3 April Kidoguchi, Kenneth

9 ( 2, 2 ) Quad II Quad III ( 3, 3 ) 5.3 Properties of Trigonometric Functions 3. Signs of the Trigonometric Functions (, ) Quad I Quad IV (r, 0) 3 April Kidoguchi, Kenneth ( 4, 4 ) sin 0, csc 0 cos 0, sec 0 tan 0, cot sin 0, csc 0 cos 0, sec 0 tan 0, cot 0 sin 0, csc 0 cos 0, sec 0 tan 0, cot sin 0, csc 0 cos 0, sec 0 tan 0, cot 0 4 2

10 Finding the Quadrant in Which an Angle Lies - Eample If sin() > 0 and cos() < 0, name the quadrant in which the angle lies. Quad II Quad I Quad III Quad IV (r, 0) 3 April Kidoguchi, Kenneth

11 Reciprocal Identities csc, sin 0 sin sec, cos 0 cos cot, tan 0 tan Quotient Identities sin tan, cos 0 cos cos cot, sin 0 sin 5.3 Properties of Trigonometric Functions Reciprocal and Quotient Identities 3 April 208 Kidoguchi, Kenneth

12 Finding Eact Values When sin() & cos() are Given - Eample Given sin 0 and cos 3 0, find the eact values of the 0 0 four remaining trigonometric functions. 3 April Kidoguchi, Kenneth

13 Pthagorean Identities For a circle of radius r, with centre at the origin of a rectangular coordinate sstem. r r r 2 2 cos sin Uncle Pthagoras Algebra sin, cos r r 2 2 cos sin Conventional notation 3 April Kidoguchi, Kenneth

14 2 2 sin cos 5.3 Properties of Trigonometric Functions Pthagorean Identities 3 April Kidoguchi, Kenneth

15 Summar of Fundamental Identities csc, sin sec, cos cot tan sin tan, cos cot cos sin 2 2 tan sec, cot 2 csc sin cos, 3 April Kidoguchi, Kenneth

16 Value of Trig Epression Using Identities - Eample Find the eact value of each epression without a calculator. 2 (a) cos 35º csc 35º 2 (b) cos 3 cot 3 sin 3 3 April Kidoguchi, Kenneth

17 5. Find the Eact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle Given that sin() = -2/3 and cos() < 0, find the eact value of each of the remaining five trigonometric functions. Quad II Quad I Quad III Quad IV 3 April Kidoguchi, Kenneth

18 5. Find the Eact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle 3 April Kidoguchi, Kenneth

19 Given One Value of a Trig Function and the Sign of Another - Eample Given that tan() = /3 and sin() < 0, find the eact value of each of the remaining five trigonometric functions. Quad II Quad I Quad III Quad IV 3 April Kidoguchi, Kenneth

20 6. Use Even-Odd Properties to Find Eact Values of the Trigonometric Functions Even-Odd Properties sin sin, cos cos, tan tan csc csc, sec sec, cot cot 3 April Kidoguchi, Kenneth

21 Find the eact values of: (a) cos(-60º) = 5.3 Properties of Trigonometric Functions Even-Odd Properties - Eample (b) sin(-390º) = (c) tan(-7/3) 3 April Kidoguchi, Kenneth