{"id":2177,"date":"2023-10-02T10:33:31","date_gmt":"2023-10-02T10:33:31","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=2177"},"modified":"2023-10-02T10:35:06","modified_gmt":"2023-10-02T10:35:06","slug":"week-8-ss1-second-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-8-ss1-second-term-further-mathematics-notes\/","title":{"rendered":"Week 8 &#8211; SS1 Second Term Further Mathematics Notes"},"content":{"rendered":"<p>\u00a0<\/p>\n<h3>WEEK  EIGHT  \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0DATE\u2026\u2026\u2026\u2026\u2026 TOPIC : TRIGONOMETRIC RATIO CONTENT<br \/>\n<\/h3>\n<ul>\n<li>Basic trigonometric Ratio<strong><br \/>\n\t\t\t\t<\/strong>\n\t\t<\/li>\n<li>Ratio of General Angle<strong><br \/>\n\t\t\t\t<\/strong>\n\t\t<\/li>\n<li>Trigonometric Identities<strong><br \/>\n\t\t\t\t<\/strong>\n\t\t<\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<h3>BASIC TRIGONOMETRIC RATIO<br \/>\n<\/h3>\n<p>The basic trigonometric ratios can be defined in terms of the sides of a right angled triangle.<br \/>\n                                    Q<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se1.png\" alt=\"\"\/>  r<br \/>\np <\/p>\n<p>\u00a0<br \/>\n\u00a0R          q                     P<br \/>\n\u25b2PQR in the figure above is a right angle triangle with QPR = \u04e8 and PRQ = 90\u02da We define the three basic ratios as follows: <\/p>\n<p>\u00a0 \u00a0\u00a0\u00a0\u00a0Cosine of angle \u04e8 =       PR<br \/>\n\t\t    q<br \/>\n\t \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0PQ   =        r<br \/>\n \u00a0\u00a0\u00a0\u00a0<br \/>\n\u00a0 \u00a0\u00a0\u00a0\u00a0Sine of angle \u04e8 =QR<br \/>\n\t\t p<br \/>\n\t PQ      =     r<br \/>\n \u00a0\u00a0\u00a0\u00a0<br \/>\n\u00a0 \u00a0\u00a0\u00a0\u00a0Tangent of angle \u04e8 = QR          p<br \/>\n \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0PR    =  r       <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se2.png\" alt=\"\"\/><\/p>\n<p>\u00a0The cosine of angle \u04e8, sine of angle \u04e8 and the tangent of angle \u04e8 will be abbreviated as  cos\u04e8, sin\u04e8 and tan\u04e8 respectively. <\/p>\n<p>\u00a0Thus:  cos\u04e8 = q       ,sin\u04e8  =      p     ,tan\u04e8 =      p <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se3.png\" alt=\"\"\/>r       r q<br \/>\nAlso,<br \/>\n \u00a0\u00a0\u00a0\u00a0sin\u04e8  =   p\/r  =   p     =  tan\u04e8<br \/>\n<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se4.png\" alt=\"\"\/><br \/>\n\tcos\u04e8      q\/r         q    <\/p>\n<p>\u00a0tan\u04e8 =  sin\u04e8<br \/>\n\tcos\u04e8 <\/p>\n<p>\u00a0<strong>Reciprocals of Basic Ratios <\/strong><br \/>\n\tWe define the reciprocals of the three basic ratios as:<br \/>\nSecant of angle \u04e8 =   PQ\/PR    = r\/q   = 1 \/ cosine of angle \u04e8.<br \/>\nCosecant of angle \u04e8 = PQ \/ QR = r \/ q = 1 \/ sine of angle \u04e8<br \/>\nCotangent of angle \u04e8 = PR \/ QR   = q \/ p =   1 \/ tangent of angle \u04e8<br \/>\nThe secant of angle \u04e8, the cosecant of angle \u04e8 and the cotangent of angle \u04e8 are abbreviated sec\u04e8, cosec\u04e8 and cot\u04e8 respectively.<br \/>\n \u00a0\u00a0\u00a0\u00a0Sec\u04e8   = r \/ q   = 1 \/ cos\u04e8<br \/>\n \u00a0\u00a0\u00a0\u00a0Cosec\u04e8   = r \/ p =   1 \/ sin\u04e8<br \/>\n \u00a0\u00a0\u00a0\u00a0Cot\u04e8 =   q \/ p =   1 \/ tan\u04e8   =   cos\u04e8 \/ sin\u04e8 <\/p>\n<p>\u00a0<\/p>\n<h4>Example 1<br \/>\n<\/h4>\n<p>Given that sin\u04e8 = 5 \/ 13 and \u04e8 is acute, find: <\/p>\n<ol>\n<li>cos\u04e8\n<\/li>\n<li>tan\u04e8\n<\/li>\n<li>sec\u04e8\n<\/li>\n<li>cosec\u04e8\n<\/li>\n<li>cot\u04e8\n<\/li>\n<\/ol>\n<p>\u00a0Solution<br \/>\n                      Q <\/p>\n<p>\u00a0<br \/>\n\u00a013<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se5.png\" alt=\"\"\/>                     5                         <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0 R        5            P<br \/>\nUse Pythagoras theorem to find PR<br \/>\n \u00a0\u00a0\u00a0\u00a0PQ<sup>2<\/sup> = PR<sup>2<\/sup> + QR<sup>2<\/sup><br \/>\n\t \u00a0\u00a0\u00a0\u00a013<sup>2<\/sup> =PR<sup>2<\/sup> + 5<sup>2<\/sup>  \u00a0\u00a0\u00a0\u00a0PR<sup>2<\/sup> = 13<sup>2<\/sup> &#8211; 5<sup>2<\/sup><br \/>\n\t \u00a0\u00a0\u00a0\u00a0       = 169 \u2013 25<br \/>\n \u00a0\u00a0\u00a0\u00a0       = 144<br \/>\n \u00a0\u00a0\u00a0\u00a0PR = 12 <\/p>\n<p>\u00a0Thus, q = 12, r = 13, p = 5. <\/p>\n<ol>\n<li>cos\u04e8 = q \/ r = 12 \/13\n<\/li>\n<li>tan\u04e8 = p \/ q = 5 \/ 12\n<\/li>\n<li>sec\u04e8 = r \/ q = 13 \/ 12\n<\/li>\n<li>cosec\u04e8 = r \/ p = 13 \/ 5\n<\/li>\n<li>cot\u04e8 = q \/ p = 12 \/ 5\n<\/li>\n<\/ol>\n<p>\u00a0<br \/>\n\t\t<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se6.png\" alt=\"\"\/><br \/>\n\tcos\u04e8= x<br \/>\ntan\u04e8 = y \/ x  <\/p>\n<p>\u00a0Example: Use table to evaluate (a) sin37 (b) cos75 (c) tan62<br \/>\nSolution<br \/>\n(a) \u00a0\u00a0\u00a0\u00a0sin37 = 0.6018 <\/p>\n<p>\t\t<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se7.png\" alt=\"\"\/><br \/>\n\tCos (180 &#8211; \u04e8) = -cos\u04e8<br \/>\nTan (180 &#8211; \u04e8) = -tan\u04e8 <\/p>\n<p>\u00a0Example: Use table to evaluate (a) sin143 (b) cos 115 (c) tan 125 Solution <\/p>\n<ol>\n<li>sin143 = sin(180-143) = sin37 = 0.6018\n<\/li>\n<li>cos115 = -cos(180-115) = -cos65 = -0.4226\n<\/li>\n<li>tan125 = -tan(180-125) = -tan55 = -1.428\n<\/li>\n<\/ol>\n<p>\u00a0<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se8.png\" alt=\"\"\/><br \/>\n\tThird Quadrant<br \/>\nSin (180 + \u04e8) =   &#8211; sin\u04e8<br \/>\nCos (180 + \u04e8) =   &#8211; cos\u04e8<br \/>\nTan (180 + \u04e8) =     tan\u04e8<br \/>\nExample: Use table to evaluate (a) sin220 (b) cos236 (c) tan242 Solution <\/p>\n<ol>\n<li>sin220 =   sin (180 + 40) =  &#8211;  sin40 =  &#8211; 0.6428\n<\/li>\n<li>cos236 =   cos (180 + 56) = &#8211; cos56 =  &#8211; 0.5992\n<\/li>\n<li>tan242 =   tan (180 + 62) =   tan 62 =   1.881\n<\/li>\n<\/ol>\n<p><img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se9.png\" alt=\"\"\/><\/p>\n<p>\u00a0Fourth Quadrant<br \/>\nSin (360 &#8211; \u04e8) =   &#8211; sin\u04e8<br \/>\nCos (360 &#8211; \u04e8) =    cos\u04e8<br \/>\nTan (360 &#8211; \u04e8) =   &#8211; tan\u04e8 <\/p>\n<p>\u00a0<br \/>\n\u00a0Example: Use table to evaluate (a) sin310<sup>0<\/sup> (b) cos285<sup>0<\/sup> (c) tan334<sup>0 <\/sup>Solution <\/p>\n<ol>\n<li>sin310<sup>0<\/sup> = &#8211; sin (360-310) =  &#8211; sin50 = &#8211; 0.7660\n<\/li>\n<li>cos285<sup>0<\/sup> =   cos (360-285)  =  cos75 =    0.2588\n<\/li>\n<li>tan334<sup>0<\/sup> = &#8211; tan (360-334) =  &#8211; tan26 =  &#8211; 0.4877\n<\/li>\n<\/ol>\n<p>\u00a0<strong>   Note that:        <\/strong><\/p>\n<ol>\n<li>In the first quadrant, all the ratios are positive.\n<\/li>\n<li>In the second quadrant, only sine ratio is positive, while the rest are negative.\n<\/li>\n<li>In the third quadrant, only tangent ratio is positive, while the rest are negative.\n<\/li>\n<li>In the fourth quadrant, only cosine ratio is positive, while the rest are negative.\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<h4>Evaluation<br \/>\n<\/h4>\n<p>1) Use tables to evaluate the following   (a) Sin 162<sup>0<\/sup>  (b) Cos 283<sup>0<\/sup>  (c) Tan 325<sup>0<\/sup>  (d) Cos( &#8211; 75)      (e)Tan (-100)   (f) Sin ( -223)        2)    Use tables to find the values \u03d5 between 0<sup>0<\/sup> and 360<sup>0<\/sup> which satisfy each of the following.<br \/>\n                (a) Sin \u03d5 = 0.4396   (b) Tan \u03d5 = &#8211; 2.4398   (c) Cos \u03d5 = 0.8427 <\/p>\n<h3>TRIGONOMETRIC IDENTITY<br \/>\n<\/h3>\n<p>Pythagoras theorem.                  Y<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se10.png\" alt=\"\"\/>P 1 y x<br \/>\n O     x     N <\/p>\n<p>\u00a0<br \/>\n\u00a0The figure above shows a unit circle. \u25b2OPN is a right angled with OP = 1, ON= x and PN = y, PON = \u04e8. From the definition of trigonometric ratios. <\/p>\n<ol>\n<li>= cos\u04e8 \u00a0\u00a0\u00a0\u00a0                   \u2026. (1)\n<\/li>\n<li>= sin\u04e8 \u00a0\u00a0\u00a0\u00a0 \u2026..(2)\n<\/li>\n<\/ol>\n<p>From (1)   x<sup>2<\/sup> = cos<sup>2<\/sup>\u04e8                          \u2026\u2026 (3)<br \/>\nFrom (2)   y<sup>2<\/sup> = sin<sup>2<\/sup>\u04e8                          \u2026\u2026 (4) Adding equations (3) and (4)  x<sup>2<\/sup> + y<sup>2<\/sup> = cos<sup>2<\/sup>\u04e8 + sin<sup>2<\/sup>\u04e8        \u2026..(5)<br \/>\nSince \u25b2OPN is a right angled triangle<br \/>\n \u00a0\u00a0\u00a0\u00a0ON<sup>2<\/sup> + NP<sup>2<\/sup> = OP<sup>2<\/sup><br \/>\n\t \u00a0\u00a0\u00a0\u00a0x<sup>2<\/sup> + y<sup>2<\/sup> = 1                          \u2026\u2026 (6) Equating equations   (5) and (6)<br \/>\n \u00a0\u00a0\u00a0\u00a0Cos<sup>2<\/sup>\u04e8 + sin<sup>2<\/sup>\u04e8 = 1               \u2026\u2026 (7)<br \/>\nDividing both sides of (7) by cos<sup>2<\/sup>\u04e8<br \/>\n \u00a0\u00a0\u00a0\u00a0Cos<sup>2<\/sup>\u04e8 \/ cos<sup>2<\/sup>\u04e8 + sin<sup>2<\/sup>\u04e8 \/ cos<sup>2<\/sup>\u04e8 = 1 \/ cos<sup>2<\/sup>\u04e8<br \/>\n \u00a0\u00a0\u00a0\u00a01 + tan<sup>2<\/sup>\u04e8 = sec<sup>2<\/sup>\u04e8               \u2026..(8)<br \/>\nDividing (7) through by sin<sup>2<\/sup>\u04e8<br \/>\n \u00a0\u00a0\u00a0\u00a0Cot<sup>2<\/sup>\u04e8 + 1 = cosec<sup>2<\/sup><br \/>\n\t \u00a0\u00a0\u00a0\u00a01 + cot<sup>2<\/sup>\u04e8 = cosec<sup>2<\/sup>\u04e8            \u2026..(9) <\/p>\n<p>\u00a0<\/p>\n<h4>Evaluation 1<br \/>\n<\/h4>\n<ol>\n<li>Prove that (1 \u2013 Sin\u03d5)(1  + Sin \u03d5)       = Cot<sup>2<\/sup>\u03d5\n<\/li>\n<\/ol>\n<p>                                Sin<sup>2<\/sup> \u03d5 <\/p>\n<ol>\n<li>Show that (Sec \u03d5 &#8211; Tan \u03d5)(Sec \u03d5  + Tan \u03d5)= 1\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<h4>Evaluation 2<br \/>\n<\/h4>\n<p>Find the values of \u0472 lying between 0 and 360 for each of the following<br \/>\n1)cos \u0472 = 0.2874 2)sin \u0472 = 0.9361<br \/>\n3)cos \u0472 =-0.8271<br \/>\n4)tan \u0472 =-2.106 <\/p>\n<p>\u00a0<\/p>\n<h3>GRAPH OF SINE AND COSINE FOR ANGLES<br \/>\n<\/h3>\n<p>In the figure below, a circle has been drawn on a Cartesian plane so that its radius, OP, is of length 1unit. Such a circle is called <strong>unit circle.<\/strong><\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0The angle \u0472 that OP makes with Ox changes according to the position of P on the circumference of the unit circle. Since P is the point (x,y) and \/OP\/ = 1 unit,<br \/>\n         Sin \u0472 = y\/1     = y<br \/>\n          Cos \u0472 = x\/1   = x Hence the values of x and y give a measure of cos \u0472 and sin \u0472 respectively.<br \/>\nIf the values of \u0472 are taken from the unit circle, they can used to draw the graph of sin \u0472. This is done by plotting values of y against corresponding values of \u0472 as in figure below. <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0In the figure above, the vertical dotted lines gives the values of sin \u0472 corresponding to \u0472 = 30, 60,90,&#8230;&#8230;., 360. To draw the graph of cos \u0472 , use corresponding values of x and \u0472. This gives another wave-shaped curve, the graph of cos \u0472 as in figure below. <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0As \u0472 increases beyond 360, both curves begin to repeat themselves as in figures below. <strong>Notice the following: <\/strong><br \/>\n\t1)All values of sin \u0472 and cos \u0472 lie between +1 and -1.<br \/>\n2)The sine and cosine curves have the same shapes but different starting points.<br \/>\n3)Each curve is symmetrical about its peak(high point) and trough(low point). This means that for any value of sin \u0472 there are usually two angles between 0 and 360; likewise cos \u0472. The only exceptions to this are at the quarter turns, where sin\u0472 and cos\u0472 have the values given in table below; <\/p>\n<div>\n<table>\n<tbody>\n<tr>\n<td> \u00a0<\/td>\n<td>0 <\/td>\n<td>90 <\/td>\n<td>180 <\/td>\n<td>270 <\/td>\n<td>360 <\/td>\n<\/tr>\n<tr>\n<td>Sin\u0472 <\/td>\n<td>0 <\/td>\n<td>1 <\/td>\n<td>0 <\/td>\n<td>-1 <\/td>\n<td>0 <\/td>\n<\/tr>\n<tr>\n<td>Cos\u0472 <\/td>\n<td>1 <\/td>\n<td>0 <\/td>\n<td>-1 <\/td>\n<td>0 <\/td>\n<td>1 <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>\u00a0<strong>Examples <\/strong><br \/>\n\t1) Reffering to graph on page 211 of NGM Book 1, (a)Find the value of sin 252, b)solve the equation   5 sin \u0472 = 4 <\/p>\n<p>\u00a0<\/p>\n<h4>Solution<br \/>\n<\/h4>\n<p>a)On the \u0472 axis, each small square represents 6. From construction a) on the graph:         Sin 252 = -0.95 b) If 5 sin \u0472 = 4    then sin \u0472 = 4\/5  = 0.8<br \/>\n  From construction (b) on the gragh: when sin \u0472 = 0.8, \u0472 = 54 or 126 <\/p>\n<p>\u00a0<\/p>\n<h4>Graph of tan \u0472<br \/>\n<\/h4>\n<p>Values can be taken from a unit circle to draw a tangent curve. In the figure below, a tangent is drawn to the unit circle OX. A typical radius is drawn and extended to meet the tangent at T. the y \u2013 coordinates of T gives a measure of tan \u0472, where \u0472 is the angle that the radius makes with OX.<br \/>\nNote that tan \u0472 is not defined when \u0472 =90<sup>0<\/sup> and 270<sup>0<\/sup>. <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<\/p>\n<h4>Ratio of special Angles (45<sup>0<\/sup>, 30<sup>0<\/sup> and 60<sup>0<\/sup>) A. Tan, Sin and Cos of 45<sup>0<\/sup><br \/>\n\t<\/h4>\n<p>Considering the diagram below; <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0\u2206ABC is right \u2013angled triangle at B and \/AB\/ = \/BC\/ = 1 unit<br \/>\n\/AC\/<sup>2<\/sup> = 1<sup>2<\/sup> + 1<sup>2<\/sup>  = 2 (by Pythagoras&#8217; theorem)<br \/>\n\/AC\/ =\u221a2<br \/>\n\tThus, tan45<sup>0<\/sup> = 1 <\/p>\n<p>\t\t<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se11.png\" alt=\"\"\/><\/p>\n<p>\t\t<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se12.png\" alt=\"\"\/><\/p>\n<p>\u00a0<\/p>\n<h4>B. Tan, Sin and Cos of 30<sup>0<\/sup> and 60<sup>0<\/sup><br \/>\n\t<\/h4>\n<p>Considering the diagram below; <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0ABC is an equilateral triangle with sides of 2 units in length. Line AD is an altitude where \/BD\/ = \/DC\/ = 1 unit. In \u2206ABD, \/AB\/<sup>2<\/sup> = \/AD\/<sup>2<\/sup> + \/BD\/<sup>2<\/sup> (by Pythagoras&#8217; theorem)<br \/>\n2<sup>2<\/sup> = \/AD\/<sup>2<\/sup> + 1<sup>2<\/sup><br \/>\n\t\/AD\/<sup>2<\/sup> = 3<br \/>\n\/AD\/ = \u221a3 units<br \/>\nSince, &lt;B = 60<sup>0<\/sup><br \/>\n\tThus, Tan 60<sup>0<\/sup> = \u221a3 \u00a0\u00a0\u00a0\u00a0 <\/p>\n<p>\t\t<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se13.png\" alt=\"\"\/><br \/>\n\t           Cos 60<sup>0 <\/sup>= \u00bd  = 0.5<sup>0<\/sup><br \/>\n\tAlso, &lt;BAD = 30<sup>0<\/sup><br \/>\n\t          Tan 30<sup>0<\/sup> = 1\/\u221a3 \u00a0\u00a0\u00a0\u00a0<br \/>\n           Sin 30<sup>0<\/sup> = \u00bd  = 0.5<sup>0<\/sup><\/p>\n<p>\t\t<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se14.png\" alt=\"\"\/><br \/>\n\t<strong>Example:<\/strong> Write the value of each the following in surd form; <\/p>\n<ol>\n<li>sin135<sup>0<\/sup>\n\t\t<\/li>\n<li>tan330<sup>0 <\/sup>\n\t\t<\/li>\n<li>cos240<sup>0<\/sup>\n\t\t<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<h4>Solution<br \/>\n<\/h4>\n<ol>\n<li>sin135<sup>0 <\/sup>= sin(180 -135) = sin 45 =  <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se15.png\" alt=\"\"\/>\n\t\t<\/li>\n<li>tan330<sup>0<\/sup> = -tan(360-330) = tan30 =   1\/\u221a3\n\t\t<\/li>\n<li>cos240<sup>0<\/sup> = cos(240 -180) = cos60 = &#8211; 1\/2\n<\/li>\n<\/ol>\n<p>\u00a0<strong>Evaluation: <\/strong><br \/>\n\t1)Using the same graph used in the above example, find the values of the following a)sin 24    b) sin 294<br \/>\n2)Use the same graph  to find the angles whose sines are as follows: a) 0.65     b)-0.15 <\/p>\n<p>\u00a0<\/p>\n<h3>GENERAL EVALUATION<br \/>\n<\/h3>\n<ol>\n<li>Use tables to evaluate each of the following (i) sin310 (ii) tan242 (iii) cos(-243) (iv) sin(-260) (iv) tan(-255)\n<\/li>\n<li>Use tables to find the values of \u0472 between 0<sup>0<\/sup> and 360<sup>0<\/sup> which satisfy each of the following (i) cos \u0472 = -0.4540 (ii) tan       \u0472= 1.176 (iii) sin \u0472 = -0.9336\n<\/li>\n<li>Using the same axis, a scale of 1cm to represent 30<sup>0<\/sup> on the \u0472-axis and 2cm to represent 1 unit on the y-axis, draw the graph of the following relations (i) y = sin \u0472 (ii) sin \u0472\/2    \u00a0\u00a0\u00a0\u00a0\n<\/li>\n<li>Simplify <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se16.png\" alt=\"\"\/>\n\t\t<\/li>\n<\/ol>\n<p>       3-  \u221a2<br \/>\n<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se17.png\" alt=\"\"\/><br \/>\n\t       3\u221a5 + \u221a2<br \/>\n<strong>READING ASSIGNMENT <\/strong><br \/>\n\tNGM BK 1 PG 187 \u2013 195; Ex 17c nos 3 and 6                                                    <\/p>\n<p>\u00a0<\/p>\n<h3>WEEKEND ASSIGNMENT<br \/>\n<\/h3>\n<p>Given that sin \u04e8 = 4\/5 and \u04e8 is acute <\/p>\n<ol>\n<li>find cos \u04e8 (a) 5\/3 (b) 3\/5 (c) 4\/3 (d) 4\/5\n<\/li>\n<li>find tan \u04e8 (a) 4\/5 (b) 3\/5 (c) 5\/4 (d) \u00be\n<\/li>\n<li>find cosec \u04e8 (a) 4\/5 (b) 3\/5 (c) 5\/4 (d) \u00be\n<\/li>\n<li>find sec \u04e8 (a) 5\/3 (b) 5\/4 (c) \u00be (d) 5\/2\n<\/li>\n<li>find cot \u04e8 (a) 3\/5 (b) 4\/5 (c) 5\/4 (d) 5\/3\n<\/li>\n<\/ol>\n<p>\u00a0<\/p>\n<h3>THEORY<\/p>\n<\/h3>\n<p>1a.)  Prove that          1            +         1             =    2 cosec<sup>2<\/sup> \u04e8<br \/>\n \u00a0\u00a0\u00a0\u00a01 + cos \u04e8  \u00a0\u00a0\u00a0\u00a01 \u2013 cos \u04e8<br \/>\nb.) Given that sin \u04e8 = 5\/ 13    and \u04e8 is acute, find (i) cos \u04e8    (ii) tan \u04e8   (iii) sec \u04e8    (iv)  cosec \u04e8    (v) cot \u04e8<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_1033_Week8SS1Se18.png\" alt=\"\"\/>2a) Copy and complete the table below, giving corresponding values of \u0472 from 0<sup>o<\/sup> to 360<sup>o<\/sup> \u0472                0     30    60    90    120    150    180    210    240    270    300    330    360 Cos \u0472         1    0.87  0.5   0    &#8211; 0.5<br \/>\nb)Hence draw the graph of cos \u0472, using 2cm to 0.5 on y-axis and 1cm to 30<sup>o<\/sup> x-axis <\/p>\n<p>\u00a0bi) Construct a table for y = cosx \u2013 3sinx for values of x from 0<sup>0<\/sup> to 180<sup>0<\/sup> at intervals of 20<sup>0<\/sup>.   ii) Using a scale of 2cm to 20<sup>0<\/sup> on the x-axis and 2cm to 1 unit on the y-axis, draw the graph of y= cosx -3sinx.   iii) Use your grah to find the value(s) of x correct to the nearest degree for which (i) 3tanx = 1(ii) 2 + cosx = 3sinx. <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<strong><br \/>\n\t\t\t<\/strong><br \/>\n\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0 WEEK EIGHT \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0DATE\u2026\u2026\u2026\u2026\u2026 TOPIC : TRIGONOMETRIC&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,188],"tags":[],"class_list":["post-2177","post","type-post","status-publish","format-standard","hentry","category-posts","category-second-term-ss1-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2177","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=2177"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2177\/revisions"}],"predecessor-version":[{"id":2178,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2177\/revisions\/2178"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=2177"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=2177"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=2177"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}