Course I. Lesson 5 Trigonometric Functions (II) 5A Radian Another Unit of Angle Graphs of Trigonometric Functions

Transcription

1 Course I Lesson 5 Trigonometric Functions (II) 5A Radian Another Unit of Angle Graphs of Trigonometric Functions

2 Radian Degree ( ) Angle of /0 of one circle 0 is a familiar number in astronom. ( ne ear = 5da 0 ) Radian (non-dimention) Angle is described b the ratio of the arc to the radius. r Arc 0 = r Radius A pure number (no unit) but smbol rad is used. rad=57.9 ( Memorize 0 = rad. ) B rad r r A

3 Merits of Radian Eample : Epression becomes simple. Area of a sector with angle and radius r [degree] : r 0 [rad] : = r r r Eample : Values of trigonometric functions of a small angle can be obtained approimatel. When angle is small BH arcab sin = = B B EX. sin = H A (From Table, deg 0.075) B

4 Graphs of the Sine/Cosine Functions r sin = cos = r r = r sin = r cos r = = sin = cos P P 4

5 Graph of the Tangent Function r tan = = = tan P 5

6 Periodic Function Periodic function A function f () is said to be periodic with period p if we have f ( + p) = f ( ) Namel, the values of a function repeat themselves regularl. Eamples Find the period of the sine functions = sin Period = +

7 r Eample Eample. Illustrate the following functions and show their periods () () () Ans. = sin () Epansion in the -direction (period =) = sin = sin = sin = sin - () Epansion in the -direction (period =) = sin = sin () Shift in the -direction (period =) = sin = sin 7

8 Eercise Eercise. Answer about the following function = sin( ) ( 0 ) () When dos this function becomes zero? () What are the values of this function at =0, () Illustrate this function. Ans. Pause the video and solve the problem. 8

9 Eercise Eercise. Answer about the following function = sin( ) ( 0 ) () When does this function become zero? () What are the values of this function at =0, () Illustrate this function. Ans. () 0, 0 4, 4 Therefore becomes zero at = 0,,, =,, () At = 0 : = sin( ) = At = : = sin(4 ) = sin( ) = () , 5 Ahh! That s so eas! 9

10 Course I Lesson 5 Trigonometric Functions (II) 5B Trigonometric Equation Trigonometric Inequalit 0

11 Trigonometric Equation A trigonometric equation is an equation that contains unknown trigonometric function. E. sin + cos = 0 This kind of equation is true for certain angles. [Note] A trigonometric equation that holds true for an angle is called a trigonometric identit, which we will stud net lesson. Some trigonometric equation can be solved easil b using algebra ideas, while others ma not be solved eactl but approimatel. Eample sin = This can be easil solved. See net slide. Eample Roots and can be found numericall (See the figure). 0 sin = 0 = sin =

12 Eample Eample. Solve the following trigonometric equation. sin = 0 Ans. Step We first look at sin solve as we did before. Step Recall the graph of the value of - 5 = sin sin = which satisf this epression. = = sin as being the variable of the equation a from 0 to or a unit circle, and obt Ⅱ Ⅲ, 5 5 Ⅰ Ⅳ Step Considering the periodicit, add n. 5 = + n, + n

13 Trigonometric Inequalit A trigonometric inequalit is an inequalit that contains unknown trigonometric function. It can be solved based on a trigonometric e Eample. Solve the following trigonometric equation. sin > 0 Ans. Step Convert the given inequalit to a trigonometric equation b replacing sign to equalit sign. sin = 0 Step Solve the resulting equation in the interval [0, ] = /, 5 / = sin = 5 Step Among intervals divided b the obtained roots, find the intervals wh 5 satisf the trigonometric inequalit. < < Step 4 Etend 5 the soltion to the whole domain. + n < < + n

14 Eercise Eercise. Solve the following trigonometric equation. sin + cos = 0 Ans. Pause the video and solve the problem. 4

15 Eercise Eercise. Solve the following trigonometric equation. sin + cos = 0 0 Ans. sin Put + cos = 0 X = cos sin + cos = From ( X ) + X X =, X = cos = = 0, = 0 X X + = 0 ( X )(X ) = 0 From cos = =,

16 Eercise Eercise. Solve the following trigonometric inequalit tan Ans. Pause the video and solve the problem.

17 Answer to the Eercise Eercise. Solve the following trigonometric inequalit tan Ans. The corresponding trigonometric equation is tan = Tangent has the period as shown in the figure. In the interval [-/, /], the root is = 4 From the graph and considering the periodicit, the solution is + n < + n 4-7 7