{"id":2849,"date":"2023-10-03T14:02:40","date_gmt":"2023-10-03T14:02:40","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=2849"},"modified":"2023-10-03T14:03:22","modified_gmt":"2023-10-03T14:03:22","slug":"week-10-ss2-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-10-ss2-first-term-further-mathematics-notes\/","title":{"rendered":"Week 10 – SS2 First Term Further Mathematics Notes"},"content":{"rendered":"

\u00a0WEEK 10
\n<\/strong>TOPIC: Trigonometric Identities and graphs of inverse trigonometric ratios
\n<\/strong>Pythagoras Theorem
\n<\/strong>\"\"\/<\/p>\n

\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 P
\n\u00a0\u00a0\u00a0\u00a0
\n\u00a0
\n\u00a0
\n\u00a0\"\"\/
\n\t\t\"\"\/\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\u00a0x<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Fig. 10 a
\n<\/strong>
\n\u00a0
\n\u00a0
\n\u00a0Fig 10a shows a unit circle \u2206OPN is a right \u2013 angled triangle with OP = 1, ON = x and PN = y, PON \u2013 \u0473. Find the definition of trigonometric ratios.
\n\u00a0\u00a0\u00a0\u00a0x = cos\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(1)
\n\u00a0\u00a0\u00a0\u00a0y = sin\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(2)
\nFrom (1)\u00a0\u00a0\u00a0\u00a0x2<\/sup> = cos2<\/sup>\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(3)<\/p>\n

\u00a0From (2)\u00a0\u00a0\u00a0\u00a0y2<\/sup> = sin2<\/sup>\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(4)<\/p>\n

\u00a0Adding (3) and (4)
\n\u00a0\u00a0\u00a0\u00a0x2<\/sup> + y2<\/sup> = cos\u04732<\/sup> + sin\u04732<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(5)<\/p>\n

\u00a0Since \u2206OPN is a right \u2013 angled triangle
\n\u00a0\u00a0\u00a0\u00a0ON2<\/sup> + NP2<\/sup> = OP2<\/sup>
\n\t\t\u00a0\u00a0\u00a0\u00a0x2<\/sup> + y2<\/sup> = 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(6)
\nEquating (5) and (6)
\ncos2<\/sup>\u0473+ sin2<\/sup>\u0473 = 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(7)<\/p>\n

\u00a0Dividing both sides of (7) by cos2<\/sup>\u0473
\ncos2<\/sup>\u0473 \u00a0\u00a0\u00a0\u00a0+ sin2<\/sup>\u0473\u00a0\u00a0\u00a0\u00a0= \u00a0\u00a0\u00a0\u00a01
\n\"\"\/\"\"\/\"\"\/cos2<\/sup>\u0473 \u00a0\u00a0\u00a0\u00a0cos2<\/sup>\u0473\u00a0\u00a0\u00a0\u00a0cos2<\/sup>\u0473
\n: 1 + tan2<\/sup>\u0473 = sec2<\/sup>\u0473
\nDividing (7) through by sin2<\/sup>\u0473
\nCos2<\/sup>\u0473 + 1 = cosec2<\/sup>\u0473
\n: 1 + cot2<\/sup>\u0473 = cosec2<\/sup>\u0473<\/p>\n

\u00a0Example
\n<\/strong>Show that (3 \u2013 sin2<\/sup>\u0473) cosec2<\/sup>\u0473 = 2cosec2<\/sup>\u0473 + cot2<\/sup>\u0473
\n<\/strong>Solution
\n<\/strong>LHS = 3cose2<\/sup>\u0473 \u2013 sin2<\/sup>\u0473 cosec2<\/sup>\u0473
\n\u00a0\u00a0\u00a0\u00a0= 3cosec2<\/sup>\u0473 \u2013 sin2<\/sup>\u0473
\n\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Sin2<\/sup>\u0473
\n\u00a0\u00a0\u00a0\u00a0= 3cosec2<\/sup>\u0473 \u2013 1
\n\u00a0\u00a0\u00a0\u00a0= 3cosec2<\/sup>\u0473 \u2013 (cosec2<\/sup>\u0473- cot2<\/sup>\u0473)
\n\u00a0\u00a0\u00a0\u00a0= 3cosec2<\/sup>\u0473 \u2013 cosec2<\/sup>\u0473 + cot2<\/sup>\u0473
\n\u00a0\u00a0\u00a0\u00a0= 2cosec2<\/sup>\u0473 + cot2<\/sup>\u0473
\n\u00a0\u00a0\u00a0\u00a0= RHS<\/p>\n

\u00a0Example
\n<\/strong>Prove that 2cosec2<\/sup>\u0473<\/p>\n

\u00a0Solution
\n<\/strong>LHS =
\n= \u00a0\u00a0\u00a0\u00a02
\n\"\"\/ 1 \u2013 cos2<\/sup>\u0473<\/p>\n

\u00a0LHS =\u00a0\u00a0\u00a0\u00a0 2
\n\"\"\/ sins2<\/sup>\u0473
\n\u00a0\u00a0\u00a0\u00a0= 2cosec2<\/sup>\u0473
\n\u00a0\u00a0\u00a0\u00a0= RHs
\nExample
\n<\/strong>Show that = 4cot2<\/sup>\u0473 + 2<\/p>\n

\u00a0Solution
\n<\/strong>LHS<\/p>\n

\u00a0= (1 + cos\u0473)2 + (1 \u2013 cos\u0473)2<\/sup>
\n\t\t\"\"\/(1 \u2013 cos\u0473)(1 + cos\u0473)
\n= 1 + 2cos\u0473 + cos2<\/sup>\u0473 + 1 \u2013 2cos\u0473 + cos2<\/sup>\u0473
\n\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 \u2013 cos2<\/sup>\u0473
\n= 2cos2<\/sup>\u0473 + 2
\n\"\"\/ Sin2<\/sup>\u0473
\n= 2(cos2<\/sup>\u0473 + 1)
\n\"\"\/ Sin2<\/sup>\u0473
\n= 2(cot2<\/sup>\u0473 + cosec2<\/sup>\u0473)
\n= 2(cot2<\/sup>\u0473 + 1 + cot2<\/sup>\u0473)
\n= 2(2cot2<\/sup>\u0473 + 1)
\n= 4cot2<\/sup>\u0473 + 2
\n= RHS<\/p>\n

\u00a0Example
\n<\/strong>If asin\u0473 + bcos\u0473 = p
\nandacos\u0473 \u2013 bsin\u0473 = q
\nshow that a2<\/sup> + b2<\/sup> = p2<\/sup> + q2<\/sup><\/p>\n

\u00a0Solution
\n<\/strong>asin\u0473 + bcos\u0473 = p\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(1)
\nacos\u0473 \u2013 bsin\u0473 = q\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(2)
\nSquaring both sides of (1)
\n\u00a0\u00a0\u00a0\u00a0(asin\u0473 + bcos)2<\/sup> = p2<\/sup>
\n\t\t: a2<\/sup> sin2<\/sup>\u0473 + 2absin\u0473 cos\u0473 + b2<\/sup>cos2<\/sup>\u0473 = p2<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(3)
\nSquaring both sides of (2)
\n(acos\u0473 \u2013 bsin\u0473)2<\/sup> = q2<\/sup>
\n\t\ta2<\/sup>cos2<\/sup>\u0473 \u2013 2absin\u0473 cos\u0473 + b2<\/sup>cos2<\/sup>\u0473 = q2<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(4)
\nAdding (3) and (4)
\na2<\/sup>sin2<\/sup>\u0473 + 2absin\u0473 cos\u0473 + b2<\/sup>cos2<\/sup>\u0473 + a2<\/sup>cos\u0473 \u2013 2absin\u0473 cos\u0473 + b2<\/sup>sin2<\/sup>\u0473 = p2<\/sup>q2<\/sup>
\n\t\t: a2<\/sup>sin2<\/sup>\u0473 + a2<\/sup>cos2<\/sup>\u0473 + b2<\/sup>cos2<\/sup>\u0473 + b2<\/sup>sin2<\/sup>\u0473 = p2<\/sup> + q2<\/sup>
\n\t\ta2<\/sup>(sin2<\/sup>\u0473 + cos2<\/sup>\u0473) + b2<\/sup>(cos2<\/sup>\u0473 + sin2<\/sup>\u0473) = p2<\/sup> + q2<\/sup>
\n\t\tBut sin2<\/sup>\u0473 + cos2<\/sup>\u0473 = 1
\n: a2<\/sup> + b2<\/sup> = p2<\/sup> + q2<\/sup><\/p>\n

\u00a0Trigonometric Equations
\n<\/strong>A trigonometric equation is an equation containing trigonometric ratios.
\nSince trigonometric functions are periodic functions, there are infinite numbers of solutions for trigonometric equations. It is therefore the tradition to find some solutions within a given range.
\nIf\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0asin\u0473 + = 0(a \u2260 0)
\nthen,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0asin\u0473 = -b
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n \u0472 = sin1<\/sup>
\n\t\tSin1<\/sup>means the angles whose sine ratio is
\nIt is called sine inverse of it is also denoted as sin thus:
\nSin\u0473 =
\n\u0472 = sin-1<\/sup><\/p>\n

\u00a0= arc sin
\nSimilarly, if acos\u0473 + b = 0, (a \u2260 0)
\nthen\u00a0\u00a0\u00a0\u00a0cos\u0473 =
\n\u0473 = cos-1<\/sup><\/p>\n

\u00a0Evaluation
\n<\/strong>Sketch the graph of:
\n<\/strong>(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 = cos x
\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

\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) Prove that sec2<\/sup>\u0473 + cosec2<\/sup>\u0473 = (tan\u0473 + cot\u0473) 2<\/sup>.<\/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","protected":false},"excerpt":{"rendered":"

\u00a0WEEK 10 TOPIC: Trigonometric Identities and graphs of inverse trigonometric ratios Pythagoras Theorem \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 P…<\/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-2849","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\/2849","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=2849"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2849\/revisions"}],"predecessor-version":[{"id":2850,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2849\/revisions\/2850"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=2849"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=2849"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=2849"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}