TRIGONOMETRIC FUNCTIONS

Transcription

1 MTF TRIGONOMETRIC FUNCTIONS NCERT Solved examples upto the section (Introduction) and (Angles) : Example : Convert into radian measure radian 0 Example : Convert radians into degree measure 0 8 approximately Example : Find the radius of the circle in which a central angle of 0 0 intercepts an arc of length cm (use ) cm Example : The minute hand of a watch is cm long How far does its tip move in 0 minutes? (Use = ) 8 cm Example : If the arcs of the same length in two circles subtend angles 0 and 0 0 at the centre, find the ratio of their radii : EXERCISE Find the radian measures corresponding to the following degree measures : (i) 0 (ii) 0 0 (iii) 0 0 (iv) 0 0 Find the degree measures corresponding to the following radian measures (Use ) (i) (ii) (iii) A wheel makes 0 revolutions in one minute Through how many radians does it turn in one second? Find the degree measure of the angle subtended at the centre of a circle of radius 00 cm by an arc of length cm (Use ) In a circle of diameter 0 cm, the length of a chord is 0 cm Find the length of minor arc of the chord If in two circles, arcs of the same length subtend angles 0 0 and 0 at the centre, find the ratio of their radii 8 Find the angle in radian through which a pendulum swings if its length is cm and the tip describes an arc of length (i) 0 cm (ii) cm (iii) cm (iv) Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaza, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 990, 8

2 MTF 9 (i) (ii) (iii) (iv) 9 (i) (ii) (iii) 00 0 (iv) : (i) (ii) (iii) NCERT Solved examples upto the section (Trigonometric Functions) : Example : If cos x = functions, x lies in the third quadrant, find the values of other five trigonometric sec x = /, sin x = /, cosec x = /, tan x = /, cot x = / Example : If cot x = functions, x lies in second quadrant, find the values of other five trogonometric tan x = /, sec x = /, cos x = /, sin x = /, cosec x = / Example 8 : Find the value of sin Example 9 : Find the value of cos ( 0 0 ) 0 EXERCISE Find the values of other five trigonometric functions in Exercises to cos x =, x lies in third quadrant sin x =, x lies in second quadrant cot x =, x lies in third quadrant sec x =, x lies in fourth quadrant tan x =, x lies in second quadrant Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaza, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 990, 8

3 MTF Find the values of the trigonometric functions in Exercises to 0 sin 0 cosec ( 0 0 ) 8 9 tan 9 sin 0 cot sinx,cosecx,secx, tanx, cotx cosecx sinx,cosx,cosecx,secx,cosx sinx,cosecx,cosx sinx,cosecx,tanx,secx,tanx,cotx,tanx,cotx,cosx,sec x,cotx NCERT Solved examples upto the section (Trigonometric Functions of Sum and Difference of Two Angles) : Example 0 : Prove that sin sec sin cot Example : Find the value of sin 0 Example : Find the value of tan Example : Prove that sin(x y) tanx tany sin(x y) tan x tan y Example : Show that tan x tan x tan x = tan x tan x tan x Example : Prove that Example : Prove that cos x cos x cosx cosx sin x sin x cot x cos x Example : Prove that sin x sin x sin x cosx cos x tan x Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaza, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 990, 8

4 MTF EXERCISE Prove that : sin cos tan sin cosec cos cot cosec tan Find the value of : (i) sin 0 (ii) tan 0 Prove the following : sin cos sec 0 cos xcos y sin xsin y sin(x y) tan x tan x tan x tan x cos( x) cos( x) 8 cot x sin( x)cos x 9 cos xcos( x) cos x cot( x) 0 sin (n + )x sin (n + )x + cos (n + )x cos (n + )x = cos x cos x cos x sin x cos x cos x = sin x sin 8x sin x + sin x + sin x = cos x sin x cot x (sin x + sin x) = cot x (sin x sin x) sin x sin x = sin x sin 0x 8 cos 9x cosx sin x sin x sin x cos0x sin x sin y x y tan cos x cosy sin x sin x tanx cosx cos x 9 sin x sin x tan x cos x cos x 0 sin x sin x sin x sin x cos x cot x cot x cot x cot x cot x cot x = cos x cos x cos x cot x sin x sin x sin x tan x( tan x) tan x cos x = 8sin x cos x tan x tan x cos x = cos x 8cos x + 8cos x (i) (ii) Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaza, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 990, 8

5 MTF NCERT Solved examples upto the section (Trigonometric Equations) : Example 8 : Find the principal solutions of the equation sin x x and Example 9 : Find the principal solutions of the equation tan x and Example 0 : Find the principal solutions of the equation sin x x n ( ) n, where n Z Example : Solve cos x x n, where n Z Example : Solve tan x cot x x n, where n Z Example : sin x sin x + sin x = 0 n x or n, where n Z Example : cos x + sin x = 0 x n ( ) n, where n Z EXERCISE Find the principal and general solutions of the following equations : tan x = sec x = cot x = cosec x = Find the general solution for each of the following equations : cos x = cos x cos x + cos x cos x = 0 sin x + cos x = 0 8 sec x = tan x 9 sin x + sin x + sin x = 0 Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaza, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 990, 8

6 MTF,,n,n Z,,n,n Z,,n,n Z,,n ( ),n Z n x or x n,n Z x (n ) or n, n Z n n n x n ( ) or (n ),n Z 8 x or,n Z 8 9 n x orn,n Z MISCELLANEOUS EXAMPLES Example : If sin (x + y) sin x, cos y, where x and y both lie in second quadrant, find the value of Example : Prove that Example : Find the value of x 9x x cos xcos cos xcos sin xsin tan 8 x x x Example 8 : If tan x =, < x <, find the value of sin,cos and tan x x x sin,cos,tan 0 0 Example 9 : Prove that cos x cos x cos x MISCELLANEOUS EXERCISE ON CHAPTER 9 cos cos cos cos 0 (sin x + sin x) sin x + (cos x cos x) cos x = 0 (cos x + cos y) + (sin x sin y) = cos x y (cos x cos y) + (sin x sin y) = sin x y sin x + sin x + sin x + sin x = cos x cos x sin x Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaza, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 990, 8

7 MTF (sin x sin x) (sin 9x sin x) tanx (cos x cos x) (cos 9x cos x) sin x + sin x sin x = sin x x x cos cos Find x x x sin,cos and tan in each of the following : 8 tan x = 9 cos x =, x in quadrant II, x in quadrant III 0 sin x =, x in quadrant II 8,, 9,, 0 8 8,, Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaza, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 990, 8