{"id":2847,"date":"2023-10-03T14:02:05","date_gmt":"2023-10-03T14:02:05","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=2847"},"modified":"2023-10-03T14:03:22","modified_gmt":"2023-10-03T14:03:22","slug":"week-9-ss2-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-9-ss2-first-term-further-mathematics-notes\/","title":{"rendered":"Week 9 – SS2 First Term Further Mathematics Notes"},"content":{"rendered":"

\u00a0
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\n\u00a0WEEK 9
\n<\/strong>TOPIC: Graphs of Trigonometric Function
\n<\/strong>The graph of the followings will be considered
\n(a) y = sin\u0473, 0\u25e6 \u2264 \u0473 \u2264 360\u25e6
\n(b) y = cos\u0473, 0\u25e6 \u2264 \u0473 \u2264 360\u25e6
\n(c) y = tan\u0473, 0\u25e6 \u2264 \u0473 \u2264 360\u25e6
\nThe graph of y = sin\u0473, \u0473\u25e6 \u2264 \u0473 \u2264 360\u25e6
\nOn the graph sheet, draw a long horizontal axis in the middle. Mark a point 0′, 3cm to the left of the origin 0. With centre 0′ draw a circle of radius 2cm. Draw a vertical axis through 0. Call the horizontal axis \u0473 \u2013 axis and the vertical axis y \u2013 axis.
\nOn the \u0473 \u2013 axis choose a scale of 2cm to 1 unit. Using your protractor, mark the angles: 0\u25e6, 30\u25e6, 60\u25e6, 90\u25e6, 120\u25e6, 150\u25e6 \u2026 330\u25e6 as shown in
\nDraw a horizontal line through 30\u25e6 on the circle. Draw a vertical line through 30\u25e6 on the circle. Draw a vertical line through 30\u25e6 on the \u0473 \u2013 axis to meet the horizontal line. Mark the point of intersection of these two lines with a small neat cross. Repeat the above procedure for the angles 60\u25e6, 90\u25e6, 120\u25e6, \u2026 330\u25e6. You will obtain a series of points. Join the points by a smooth curve. The curve you obtain is the graph of y = sin\u0473.<\/p>\n

\u00a0Essential features of the graph of y = sin\u0473:
\n(a) The graph of y = sin\u0473 forms a wave \u2013 like pattern. It is said to oscillate.
\n<\/strong>(b) The maximum value of y = sin\u0473 is 1 and it occurs when \u0473 = 90\u25e6.
\n(c) The minimum value of y = sin\u0473 is -1 it occurs when \u0473 = 270\u25e6.
\n(d) The graph repeats itself at intervals of 360\u25e6. The sine function is an example of a periodic function because it repeats itself at intervals of 360\u25e6. The function is said to have a periodicity<\/strong> of 360\u25e6.
\n(e) The length AE<\/em> on the graph is called the amplitude<\/strong> of the function.<\/p>\n

\u00a0The Graph of y = cos\u0473, \u0473\u25e6 \u2264 \u0473 \u2264 360\u25e6
\n<\/strong>The graph of y = cos\u0473 can be drawn in a manner similar to that of y = sin\u0473 except that the angles are measured from OR<\/em><\/strong> in the clockwise sense as shown in Fig. 14.17. This is so because cos\u0473 = sin(90\u25e6 – \u0473).
\nEssential features of the graph of y = cos\u0473:
\n(a) All the essential features that hold for the graph of y = sin\u0473 also hold for the graph of y = cos\u0473.
\n(b) In addition, the cosine curve lags behind the sine curve by a difference of 90\u25e6. The difference is usually called a Phase difference.<\/strong> In other words, the cosine curve lags behind the sine curve by a phase difference of 90\u25e6.
\n(c) Both the sine curve and the cosine curve demonstrate some physical phenomena like tidal waves, sound waves alternating currents, e.t.c.<\/p>\n

\u00a0The Graph of y = tan\u0473, \u0473\u25e6 \u2264 360\u25e6<\/strong>
\n\t\tThe graph of y = tan\u0473 is easier to draw using a table of values than using projections from a unit circle. Make a table of values of y = tan\u0473 from 0\u25e6 to 360\u25e6 as shown in Table 14.2
\nEssential features of the tangent curve:
\n(a) The curve consists of three parts between 0\u25e6 and 360\u25e6.
\n(b) Since the tangent function is not defined at 90\u25e6 and 270\u25e6, the function is said to be discontinuous <\/strong>at these points.(c) The curve rises and falls rapidly at anglesvery close to 90\u25e6 and 270\u25e6 respectively. The curve approaches the vertical lines at 90\u25e6 and 270\u25e6 but never touches them. These vertical lines at 90\u25e6 and 270\u25e6 ate called Asymptotes<\/strong>. The asymptotes are shown by dotted lines.<\/p>\n

\u00a0(d) The tangent function is also a periodic function. It has a periodicity of 180\u25e6.<\/p>\n

\n\n\n\n\n
0<\/td>\n0\u25e6<\/td>\n30\u25e6<\/td>\n60\u25e6<\/td>\n75\u25e6<\/td>\n105\u25e6<\/td>\n120\u25e6<\/td>\n150\u25e6<\/td>\n180\u25e6<\/td>\n210\u25e6<\/td>\n240\u25e6<\/td>\n255\u25e6<\/td>\n285\u25e6<\/td>\n315\u25e6<\/td>\n330\u25e6<\/td>\n300\u25e6<\/td>\n360\u25e6<\/td>\n<\/tr>\n
y = tan\u0473<\/td>\n0<\/td>\n0.58<\/td>\n1.73<\/td>\n3.73<\/td>\n-3.73<\/td>\n-1.73<\/td>\n-0.58<\/td>\n0<\/td>\n0.58<\/td>\n1.73<\/td>\n3.73<\/td>\n-3.73<\/td>\n-1<\/td>\n0.58<\/td>\n-1.73<\/td>\n0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Take a scale of 1cm to represent 30\u25e6 on the \u0473- axis and 1cm to represent 1 unit on the y \u2013 axis.<\/p>\n

\u00a0Example
\n<\/strong>Using the same axis, a scale of 1cm to represent 30\u25e6 on the \u0473 axis and 2cm to represent 1 unit on the y-axis, draw the graphs of the following relations.
\n(a) y = sin\u0473
\n(b) y = 2sin\u0473
\n(c)y = \u00bd sin\u0473 in the interval 0\u25e6 \u2264 \u0473 \u2264 360\u25e6.<\/p>\n

\u00a0Solution
\n<\/strong>(Refer to table 14.3 and Fig. 14.19)
\nWe observe that the curves:
\ny = sin\u0473
\ny = 2sin\u0473
\ny = \u00bd sin\u0473
\nhave the same periodicity (360\u25e6), but differ in amplitudes.The amplitude of y = sin\u0473 is 1.
\nThe amplitude of y = 2sin\u0473 is 2.
\nThe amplitude of y = \u00bd sin\u0473is \u00bd.
\nIn general, the curve y = A<\/em>sin\u0473 has amplitude \/A<\/em>\/ abd a periodicity of 360\u25e6. This property of the same curve is also a characteristic of the cosine curve.<\/p>\n

\u00a0Evaluation
\n<\/strong><\/p>\n

    \n
  1. Prove that sec2<\/sup>\u0473 + cosec2<\/sup>\u0473 = (tan\u0473 + cot\u0473) 2<\/sup>.\n<\/li>\n<\/ol>\n

    \u00a0General Evaluation
    \n<\/strong>(1) Draw the graph of y = 2cosx \u2013 1 in the range 0\u25e6 \u2264 x \u2264 360\u25e6 at intervals of 30\u25e6.
    \n(2) Draw the graph of y = 3sin x \u2013 1 in the range of 0\u25e6 \u2264 x \u2264 360\u25e6 at intervals of 30\u25e6
    \n(3) Sketch the graph of:
    \n(i) y = sin2x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(ii) y = cosx
    \n(iii) y = sec x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(iv) cosec x
    \nall at intervals of 30\u25e6 range 0\u2264 x \u2264 360.<\/p>\n

    \u00a0Weekend Assignment
    \n<\/strong>Given that 4cos x + 3sin x = 5, find the value of
    \n(1) Sin x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n(2) Cos x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n(3) Tan \u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n(4) Cot \u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n(5) Sin x + cos x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0Theory
    \n<\/strong>(1) Draw the graph of inverse trig function for sin x \u2013
    \n(2) Find the inverse of the following and their domains
    \n(a) y = sin x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) y = cos x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(c) y = tan x
    \n(d) y = cosec x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(e) y = sec x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(e) y = cot x<\/p>\n

    \u00a0
    \n\u00a0
    \n\u00a0
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    \n\u00a0
    \n\u00a0
    \n\t\t\t<\/strong>\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0WEEK 9 TOPIC: Graphs of Trigonometric Function The graph…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,230],"tags":[],"class_list":["post-2847","post","type-post","status-publish","format-standard","hentry","category-posts","category-first-term-ss2-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2847","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=2847"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2847\/revisions"}],"predecessor-version":[{"id":2848,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2847\/revisions\/2848"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=2847"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=2847"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=2847"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}