II. Trigonometric Function<\/span>
\nConsider a unit circle with centre at the origin of the coordinate axis.
\nLet P (a, b) be any point on the circle which makes an angle \u03b8\u00b0 with the x-axis. Let x be the corresponding radian measure of the angle \u03b8\u00b0, i.e; x is the arc length corresponding to \u03b8\u00b0.<\/p>\n<\/p>\n
From the \u2206OMP’m the figure we get;
\nsin \u03b8 = sin x = \\(\\frac{b}{1}\\) = b and cos \u03b8 = cos x = \\(\\frac{b}{1}\\) = a
\nThis means that for each real value of x we get corresponding unique \u2018sin\u2019 and \u2018cosine\u2019 value which is also real. Hence we can define the six trigonometric functions as follows.<\/p>\n
1. f : R \u2192 [-1, 1] defined by f(x) = sin x
\n<\/p>\n
2. f : R \u2192 [-1, 1] defined by f(x) = cos x
\n<\/p>\n
3. f : R – {n\u03c0, n \u2208 Z} \u2192 R – (-1, 1) defined by f(x) = \\(\\frac{1}{\\sin x}\\) = cosec x
\n<\/p>\n
4. f : R – {(2n + 1) \\(\\frac{\\pi}{2}\\)} \u2192 R – (-1, 1) defined by f(x) = \\(\\frac{1}{\\cos x}\\) = sec x
\n<\/p>\n
5. f : R – {(2n + 1)\u03c0, n \u2208 Z} \u2192 R defined by f(x) = \\(\\frac{\\sin x}{\\cos x}\\) = tan x
\n<\/p>\n
6. f : R – {n\u03c0, n \u2208 Z} \u2192 R defined by f(x) = \\(\\frac{\\cos x}{\\sin x}\\) = cot x
\n<\/p>\n
Sign of trigonometric functions in different quadrants;
\n
\nFor odd multiple of \\(\\frac{\\pi}{2}\\) trignometric functions changes as given below.
\nsin \u2192 cos
\ncos \u2192 sin
\nsec \u2192 cosec
\ncosec \u2192 sec
\ntan \u2192 cot
\ncot \u2192 tan<\/p>\n
The value of trigonometric functions for some specific angles;
\n<\/p>\n
III. Compound Angle Formula<\/span><\/p>\nsin(x + y) = sin x cos y + cos x sin y<\/p>\n
sin(x – y) = sin x cos y – cos x sin y<\/p>\n
cos(x + y) = cos x cos y – sin x sin y<\/p>\n
cos(x – y) = cos x cos y + sin x sin y
\n
\nsin(x + y) sin(x – y) = sin2<\/sup> x – sin2<\/sup> y = cos2<\/sup> x – cos2<\/sup> y<\/p>\ncos(x + y) cos(x – y) = cos2<\/sup> x – sin2<\/sup> y
\n<\/p>\nIV. Multiple Angle Formula<\/span><\/p>\ncos2x = cos2<\/sup> x – sin2<\/sup> x
\n= 1 – 2sin2<\/sup> x
\n= 2 cos2<\/sup> x – 1
\n= \\(\\frac{1-\\tan ^{2} x}{1+\\tan ^{2} x}\\)<\/p>\n<\/p>\n
V. Sub-Multiple Angle Formula<\/span>
\n<\/p>\n<\/p>\n
VI. Sum Formula<\/span>
\n<\/p>\nVII. Product Formula<\/span><\/p>\n2 sin x cos y = sin(x + y) + sin(x – y)<\/p>\n
2 cos x sin y = sin(x + y) – sin(x – y)<\/p>\n
2 cos x cos y = cos(x + y) + cos(x – y)<\/p>\n
2 sin x sin y = cos(x – y) – cos(x + y)<\/p>\n
VIII. Solution of Trigonometric Equations<\/span><\/p>\nsin x = 0 gives x = n\u03c0, where n \u2208 Z<\/p>\n
cos x = 0 gives x = (2n + 1)\u03c0, where n \u2208 Z<\/p>\n
tanx = 0 gives x = n\u03c0, where n \u2208 Z<\/p>\n
sin x = sin y \u21d2 x = n\u03c0 + (-1)n<\/sup> y, where n \u2208 Z<\/p>\ncos x = cos y \u21d2 x = 2n\u03c0 \u00b1 y, where n \u2208 Z<\/p>\n
tan x = tan y \u21d2 x = n\u03c0 + y, where n \u2208 Z<\/p>\n
Principal solution is the solution which lies in the interval 0 \u2264 x \u2264 2\u03c0.<\/p>\n
IX. Sine and Cosine formulae<\/span><\/p>\nLet ABC be a triangle. By angle A we mean the angle between the sides AB and AC which lies between 0\u00b0 and 180\u00b0. The angles B and C are similarly defined. The sides AB, BC, and CA opposite to the vertices C, A, and B will be denoted by c, a, and b, respectively.<\/p>\n
Theorem 1 (sine formula): In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC
\n\\(\\frac{\\sin A}{a}=\\frac{\\sin B}{b}=\\frac{\\sin C}{c}\\)<\/p>\n
Theorem 2 (Cosine formulae): Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B, and C, respectively, then
\na2<\/sup> = b2<\/sup> + c2<\/sup> – 2bc cos A
\nb2<\/sup> = c2<\/sup> + a2<\/sup> – 2ca cos B
\nc2<\/sup> = a2<\/sup> + b2<\/sup> – 2ab cos C<\/p>\nA convenient form of the cosine formulae, when angles are to be found are as follows:
\n<\/p>\n
We hope the Plus One Maths Notes Chapter 3 Trigonometric Functions help you. If you have any query regarding Kerala Plus One Maths Notes Chapter 3 Trigonometric Functions, drop a comment below and we will get back to you at the earliest.<\/p>\n","protected":false},"excerpt":{"rendered":"
Plus One Maths Notes Chapter 3 Trigonometric Functions is part of Plus One Maths Notes. Here we have given Kerala Plus One Maths Notes Chapter 3 Trigonometric Functions. Board SCERT, Kerala Text Book NCERT Based Class Plus One Subject Maths Notes Chapter Chapter 3 Chapter Name Trigonometric Functions Category Plus\u00a0One Kerala Kerala Plus One Maths […]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[42728],"tags":[],"yoast_head":"\n
Plus One Maths Notes Chapter 3 Trigonometric Functions - A Plus Topper<\/title>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\n\t\n