# BASICS OF TRIGONOMETRY - PART 1 - 10 TH MATHS TUTORIAL - SSC/ICSE/CBSE CLASSES - FORMULAS https://www.youtube.com/watch?v=Mv1HZYcEGpQ [00:05] Hi. [00:06] I'm Rajesh MSU Mathematics. [00:08] Welcome to Rectv Education. [00:11] Today we are discussing the topic trigonometry. [00:24] That is a very important trigonometry. [00:26] That is a very important branch of mathematics. [00:28] It deals with the angles and the sides of triangles. [00:31] So the word trigonometry is came from Greek language. [00:34] So this is the combination of three words. [00:38] First one three. [00:40] T right tri that means three. [00:41] T h r e e. [00:44] So here the second word is gonia. [00:47] G o n i a. [00:51] So that is the meaning angle. [00:54] And the next one the last one is meta metric matrix is nothing but metron. [00:57] So that is measurement. [01:02] Now the trigonometry is the combination of three words. [01:05] Those three words are [01:06] greek words uh trigonia and metric so that is a three angle measurement the topic which deals with three angles so that is triangles so the in trigonometry the first topic is trigonometric ratios [01:25] like in english alphabet english language we have the alphabets abcd like that in trigonometry we have six trigonometric ratios so those are the basic words [01:34] first one sign next cos tan cosecant secant and quarter so these six we have six trigonometric ratios so these are the ratios of the sides of triangles [01:50] first you have to consider any right angle triangle [01:53] so this is a b c this is a right angle triangle [01:56] angle b is equal to 90 degrees in this triangle if this angle is equal to theta now this side becomes exact opposite side of [02:08] angle theta and the side which is exactly opposite to higher 90 degrees that is called hypotenuse. [02:19] the remaining side that is adjacent side to this angle that is called adjacent side of angle theta. [02:30] now sine theta is defined as the we are applying this sine function for this theta sine theta is not the multiplication of sine and theta it is a completely one single function. [02:42] sine theta that is the ratio of opposite side to the hypotenuse so here sine theta is equal to opposite side of angle theta divided by hypotenuse. [03:03] so here we are calling this as a trigonometric ratio so this is the ratio to the opposite opposite side of angle. [03:10] theta to the hypotenuse. [03:12] so next trigonometric ratio is cos theta. [03:15] cos theta is equal first we finish the opposite side and next uh the second one is the adjacent side. [03:21] so now cos theta is equal to adjacent side of angle theta divided by hypotenuse. [03:32] so this is second one third one is in the third one we have to eliminate this hypotenuse. [03:37] so the tan theta is the ratio of supposed side and the hypotenuse. [03:45] yes opposite side of angle theta divided by adjacent side of angle theta. [03:57] so these are the first three trigonometric ratios. [04:00] next uh these three trigonometric ratios are the reciprocals of first these three ratios. [04:07] so that means cosecant theta cosine theta cyc cosec theta. [04:11] is equal to for this reciprocal that means opposite side of angle theta divided by hypotenuse is a sine theta cosec theta is hypotenuse divided by hypotenuse divided by opposite side of angle theta and next second one is in this sec theta is c c theta so that is the reciprocal of cosecant cos theta now sec theta is equal to hypotenuse divided by hypotenuse divided by adjacent side of angle theta now the last one is cos theta cos theta is equal to that is reciprocal of tan theta so that is equal to adjacent side of angle theta divided by opposite side of angle theta so here all these are the [05:13] ratios of different sides here so these three are the basic uh trigonometric ratios. [05:20] six sine theta what is the full name of this one? [05:24] so s i n e sine of angle theta. [05:28] so here that is the full form s i n e. [05:31] sine is the full form of sine and next one cosines. [05:35] cos c o s cos theta cos theta that is full name is cosine c o s i n equals sine. [05:42] cosine of angle theta. [05:46] the next one tan is there tan theta tan is nothing but tangent tng nt tangent of angle theta. [05:53] you can write like this and the next one cosecant theta cosec theta full name is cosecant theta c o s e c a n t cosecant of angle theta. [06:06] and sec theta sec theta is secant a c c a n t secant of angle theta. [06:14] next last one that is a cot theta co theta is cotangent. [06:21] cotangent of angle theta so these are the full forms of these six trigonometric ratios uh trigonometric identities. [06:35] trigonometric identities we have three trigonometric identities so they every day in trigonometry we are using these trigonometric identities so very important concept. [06:48] and we have three important identities first identity is sine squared theta plus cos square theta is always one our next value. [06:58] next identity secant square theta minus tan square theta is equal to 1 and the last and third identity is cosecant square theta minus cos square theta is equal to 1. [07:13] these are three identities so here [07:16] first we are proving this one sine square theta plus cos square theta is equal to one [07:20] and you consider a right angle triangle [07:25] so right angle triangle abc here the angle is equal to 90 degrees [07:34] in this trigonometry so for every question we are taking right angle triangles [07:39] so in this also we are taking abc that is a right angle triangle and angle b is equal to 90 degrees [07:44] and you consider here the angle is equal to some theta [07:48] from this right angle triangle let triangle abc is right angled at b and angle b is equal to 90 degrees and angle a is equal to theta [08:07] from this triangle abc we can write sine theta is equal to [08:16] opposite side divided by hypotenuse what is opposite side bc divided by hypotenuse ac and cos theta is equal to what is adjacent side so here that is a b a b divided by ac [08:31] so in right angle triangle by using pythagoras theorem by using pythagoras theorem hypotenuse whole square that is ac square is equal to side 1 square vc square plus ps square [08:51] so here for sine theta and cos theta the denominator is ac you have to remember that one for this expression you have to divide both lhs and rhs by ac square [09:05] so ac square divided by ac square is equal to bc square plus ba square divided by ac square a6 by ac square is equal to 1 [09:16] you have to split decompose at this plus [09:19] one. [09:19] So bc square by ac square plus ba square divided by ac square. [09:26] So here one is equal to bc by ac whole square you can write this one so whole square plus ba divided by ac whole square just you observe that one. [09:37] bc by ac is equal to sine theta and a b by ac is equal to cos theta. [09:42] In place of these two expressions you have to substitute these two values. [09:46] So implies bc by ac means sine theta whole square plus cos theta whole square is equal to 1. [09:53] Implies sine square theta plus cos square theta is equal to 1 is the first trigonometric identity. [10:05] sine theta and cosecant theta are reciprocals. [10:13] So now we are discussing about reciprocals in trigonometric ratio so here sine. [10:20] theta is equal to 1 by cosecant theta. [10:24] and next one cosecant theta is equal to 1 by sine theta. [10:30] from this 2 we can write that cosecant sine theta into cosecant theta is equal to 1. [10:37] and next cos theta and secant theta are reciprocals. [10:47] so in this cos theta is equal to 1 by secant theta. [10:50] and secant theta is equal to 1 by cos theta. [10:57] from this you can write cos theta into secant theta is equal to 1. [11:00] so this is the second result if you consider the ratio sine theta divided by cos theta. [11:08] sine theta y sine theta opposite side of angle theta divided by hypotenuse. [11:17] and the second what is cos theta. [11:19] adjacent [11:20] side of angle theta divided by hypotenuse so this is the ratio now we are considering this ratio if you cancel these two hypotenuse and hypotenuse [11:30] the next one next step opposite side of angle theta divided by adjacent side of angle theta so that is nothing but tan theta [11:45] so from this we can write so sine theta divided by cos theta is equal to tan theta [11:52] similarly we can write cos theta divided by sine theta is equal to cos theta [12:00] these two are the important results trigonometric third identity the last one so that is cosecant square theta minus cos square theta is equal to 1 [12:12] again we have to take triangle abc right angle triangle and angle b is equal to 90 degrees angle a is equal to theta [12:18] from this first we have to find cosecant theta what is cosecant theta [12:21] hypotenuse divided by opposite side what is hypotenuse. [12:25] ac divided by opposite side bc. [12:28] our next card theta cos theta is equal to adjacent side divided by opposite side. [12:34] what is adjacent side. [12:35] a b divided by opposite side bc. [12:40] so now by using pythagoras theorem. [12:46] by using pythagoras theorem is hypotenuse whole square is equal to side 1 square plus side 2 square. [12:58] so here for these two the denominator is bc you have to divide the total expression by bc square on both sides. [13:03] implies ac square divided by bc square is equal to bc square plus a b square divided by bc square. [13:14] implies ac square by bc square is equal to split at this plus so bc square by bc square plus a b square divided by bc square now. [13:24] bc square bc square both the values are same [13:27] value 1 implies now this plus value we have to bring this side and that becomes negative value so [13:34] ac by bc whole square minus a b by bc whole square is equal to 1 [13:41] just ac by bc what is this value cosecant theta you have to substitute this cosecant theta [13:47] this value here implies ac by bc that is equal to cosecant theta [13:53] cosecant theta whole square minus a b by bc [13:56] a b by bc that value is equal to cos theta [14:00] cos theta whole square that equal to 1 implies [14:02] cosecant square theta minus cos square theta is equal to 1 so [14:06] this is trigonometric third identity [14:11] an example question on this trigonometric ratios so in triangle abc in right angle triangle abc [14:25] if angle b is equal to 90 degrees [14:28] and sine theta sine b [14:32] sine c is equal to 3 by [14:35] 5 then find [14:38] all other trigonometric [14:42] ratios this is our question [14:45] on trigonometric ratios first we have to [14:48] consider [14:49] a right angle triangle all the [14:51] trigonometric ratios are defined [14:53] on only right angle triangle so here [14:57] this is a right angle [14:58] and angle abc we have to find [15:02] ratios at this angle c [15:06] so angle b is equal to 90 degrees and [15:09] sine c is equal to 3 by 5 so [15:13] what is sine c from this triangle [15:16] sine c is equal to formula opposite side [15:19] divided by hypotenuse what is opposite [15:23] side for this theta [15:24] or this opposite side a b that is equal [15:26] to a b divided by hypotenuse is ac. [15:30] so implies sine c is equal to a b divided by ac. [15:34] so here sine c you keep the value here a b by ac is equal to 3 by 5. [15:42] so that means this is a ratio you have to take the values if k is the common factor so that is 3 k by 5 k. [15:49] now by comparing these two a b you can take as a b is equal to 3 k and ac is equal to 5 k. [15:59] these are the lengths of the sides we don't know the value of k. [16:03] we have to eliminate this k from the right angle triangle from right angle triangle abc by using pythagoras theorem. [16:20] by using pythagoras theorem hypotenuse whole square is equal to sum of the squares of the two sides. [16:27] so that means from this you can write ac [16:29] ac square is equal to a b square plus bc square [16:33] here ac is the hypotenuse a b is the first side and the bc is the second side [16:40] how to find this bc value [16:42] so that means ac square what is that value [16:45] phi k whole square is equal to a b square value 3 k whole square plus bc square [16:50] we don't know that value how to find the 12 vc square [16:54] implies so 5 k whole square means 25 k square [16:57] is equal to 3 k whole square means 9 k square plus bc square [17:01] implies bc square is equal to 25 k square minus 9 k square that is equal to 16 k square [17:10] bc square is equal to this one implies bc is equal to 4k [17:14] so square root of 16 k square [17:17] from this you can get this value is equal to 4k [17:21] now already you know that sine c is equal to what is that [17:25] sine c opposite side by hypotenuse what [17:28] is opposite side [17:28] a b divided by ac that is equal to 3k [17:32] divided by phi k so that is 3 by 5 [17:35] this is a given value [17:36] now cos c is equal to adjacent side [17:39] divided by hypotenuse adjacent side is nothing but bc [17:43] by hypotenuse is ac what is bc [17:46] just now we got that value bc is equal to 4k [17:49] 4k divided by ac value 5k kk will go [17:52] that is equal to 4 by 5 [17:55] and next we know that already tan c is equal to [17:59] opposite side divided by adjacent side [18:02] opposite side a b divided by bc a b [18:05] is equal to 3 k by bc is equal to 4k kk [18:09] will go [18:09] that is equal to 3 by 4 and next [18:12] cosecant c is nothing but one by sine c [18:15] you can write directly so here three by five [18:17] for that you write reciprocal five by three [18:21] and next one secant c that is a reciprocal of cos c [18:23] cos c is equal to just four by five [18:28] for secant c you have to reverse this one so that is five by four and next chord c chord c is equal to one by ten c that is equal to one by what is ten c three by four just you write here three by four that is equal to four by three so therefore these are the six trigonometric ratios for this triangle example two if first one in triangle abc if angle b is equal to 90 degrees and tan a is equal to 5 by 12 then fine the value of cos square theta plus sine square theta divided by secant square theta minus tan square theta just ok answer [19:29] just you consider a triangle right angle triangle. [19:33] so here angle b he gave that angle is equal to 90 degrees. [19:37] and tan you give that's why you take a here and see. [19:40] this one tanya formula tan a is equal to opposite side of angle a divided by adjacent side of angle a. [19:52] what is opposite side of this angle a bc. [19:55] bc divided by adjacent side of anglia is a b. [19:59] b c divided by a b how to take. [20:03] now here the tang a is equal to 5 by 12. [20:07] so that is equal to what is tan a b c by a b. [20:11] b c by a b that is equal to 5 by 12. [20:15] just you take a common factor so that is k. [20:19] so divided by k again so that is equal to phi k divided by 12 k. [20:24] now consider the side bc length is equal to phi k. [20:28] and a b length is equal to 12 k. [20:32] so as a example one here also you have [20:36] to find this side so you know this side [20:37] this side and you don't know this one [20:39] you have to find that ac [20:41] by using pythagoras from triangle abc [20:44] that is a right angle triangle [20:46] by using pythagoras theorem [20:53] ac square [20:55] by using pythagoras theorem ac square [20:56] hypotenuse whole square [20:57] is equal to side 1 square that means bc [20:59] square plus side 2 square means ba [21:01] square [21:02] so bc what is bc of 5k just you [21:05] do this one phi k whole square plus what [21:08] is a b of [21:09] a that is equal to 12 k whole square [21:11] that is equal to [21:12] 25 k square plus 144 k square [21:16] that value is equal to 169 k square [21:19] now ac square is equal to 169 k [21:22] square implies ac is equal to square [21:25] root of 169 [21:27] k square that is equal to 13 13 169 [21:30] just to write 13k so implies ac is equal [21:33] to [21:33] 13k now you got all the [21:37] values so you know that's one a b value [21:40] bc value and ca value from this you have [21:43] to find [21:44] cos sine secant and substitute those [21:47] values [21:48] in this expression [21:53] so here the angle here just you consider [21:55] this as theta [21:57] from this triangle you know you can [21:59] write this sign [22:00] first we have to find cos theta cos [22:02] theta is equal to what is this value cos [22:04] theta cos theta is equal to adjacent [22:06] side divided by hypotenuse [22:08] adjacent side va divided by ac what is [22:10] va ba [22:11] 12k divided by 13k so here kk will go [22:14] and that big value becomes 12 by 13 [22:17] next value sine theta sine theta is [22:19] equal to opposite side divided by [22:22] hypotenuse what is opposite side so here [22:24] bc bc divided by [22:26] ac bc is equal to 5k and ac value 13k [22:29] kk will go and this value is a 5 by 13. [22:34] now you got sine theta and cos theta and [22:36] next we have to find one more value that [22:38] is secant theta [22:40] so secant theta is equal to 1 by cos [22:43] theta [22:43] you can write this one that is equal to [22:45] 1 by 12 by 13 [22:47] just you write the reciprocal 13 by [22:51] 2 already tan data that is the given [22:53] value so tan theta is equal to what is [22:55] that value [22:56] for given value 5 by [22:59] 12 now the given expression is [23:03] cos square theta plus sine square theta [23:06] divided by [23:07] secant square theta minus tan square [23:10] theta [23:10] so here just your cos square theta [23:14] is nothing but cos theta whole square [23:20] what is cos theta 12 by 13 [23:23] 12 by 13 whole square plus sine theta [23:26] value [23:26] 5 by 13 whole square divided by just you [23:30] see here secant theta [23:31] 13 by 12 whole square minus [23:35] theta value 5 by 12 whole square just [23:37] you have to substitute all the values in [23:39] the given expression [23:40] now just um 12 square means 144 by [23:43] 169 plus 25 by 169 [23:47] divided by 13 square 169 by [23:50] 12 square means 144 and minus [23:54] 5 square 25 by 12 square 144 this is [23:57] numerator and this is denominator so [23:58] here [23:59] what is the denominator is same for [24:01] these two you can add [24:02] 169 divided by 169 that is numerator by [24:07] 144 divided by 144 you will get this [24:10] one time and this one so 1 by 1 that [24:13] equal to 1 [24:14] therefore cos square theta plus sine [24:17] square theta [24:18] divided by secant square theta minus tan [24:21] square theta the value is equal to [24:24] [Music] [24:28] 1.