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Notes On Trigonometric Formula

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Published in: Mathematics
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All trigonometric formula basic to high

Taher A / Guwahati

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  1. Right rm•iangle Definition Assume that: or oo < O < 900 5 hypotenuse opposite Trigonometric Formula Sheet Definition of the Trig Functions —1 —1 Unit Circle Definition Assume 0 can be any angle. 1 adjacent opp sin 0 hyp adj cos 0 hyp opp tan 0 adj hyp CSC 0 opp hyp sec 0 adj adj cot 0 opp Domains of the sin 0 1 cos 0 1 tan 0 Trig Functions 1 csc 0 1 sec 0 cot 0 sin 0, cos 0, tan 0, YO e (—00, 00 YO e (—00, 00 V 0 # (n + — T, where n e Z Ranges of the —1 < sino < 1 —1 < cos0 < 1 —oo < tano < oo Periods of the csc 0, V 0 # nn, where n e Z sec 0, V 0 # (n + — T, where n e Z cot 0, V 0 # nn, where n e Z Trig Functions csc0 > 1 and csc0 < seco > 1 and seco < —oo < cot 0 < oo Trig Functions The period of a function is the number, T, such that f (0 +T ) — f (0 ) . So, if w is a fixed number and 0 is any angle we have the following periods. Sin(wo) -9 T w 27T cos(w0) T w tan(w0) T w csc(w0) sec(w0) cot(w0) 1 w 27T w w
  2. sin 0 tan 0 cos 0 Identities Tangent and Cotangent Identities cot(—0) — — cot 0 and Formulas Half Angle Formulas cos 0 cot 0 sin 0 Reciprocal Identities 1 sin 0 CSC 0 1 cos 0 sec 0 1 tan 0 cot 0 1 CSC 0 sin 0 1 sec 0 cos 0 1 cot 0 tan 0 sin 0 cos 0 tan 0 1 — cos(20) 2 I cos(20) 2 1 — cos(20) I cos(20) Pythagorean Identities sin2 0 + cos2 0 —1 tan2 0 + 1 — sec2 0 1 + cot2 0 — csc2 0 Even and Odd Formulas sin(—0) — sin 0 cos(—0) — cos 0 tan(—0) — — tan 0 Periodic Formulas If n is an integer sin(0 + 2nn) — sin 0 cos(0 + 2nn) — cos 0 tan(0 + an) — tan 0 csc(—0) sec (— o) — CSC 0 — sec 0 csc(0 + '2nn) sec(0 + 2nn) cot(0 + an) — CSC 0 — sec 0 — cot 0 Sum and Difference Formulas sin(a ± {3) — sin a cosß ± cos a sin 13 cos(a ± [3) — cos a cosß sin a sin 13 tan a ± tan {3 tan(a ± [3) 1 tan a tan [3 Product to Sum Formulas 1 [cos a — [3) — cos(a + sin a sin {3 2 1 [cos a — (3) + cos(a + cos a cos 13 2 1 — [sin(a + [3) + sin(a — sin a cos [3 2 1 [sin(a + (3) — sin a — cos a sin" — 2 Sum to Product Formulas Double Angle Formulas sin(20) cos(20) tan(20) — 2 sin 0 cos 0 — cos2 0 — sin — 2 cos2 0 — 1 — 1 — 2 sin2 0 2 tan 0 1 — tan2 0 sin a + sin" — 2 sin sin a — sin" — 2 cos cos a + cos {3 — 2 cos cos a — cosß — —2 sin Cofunction cos 2 sin 2 cos 2 sin 2 2 2 Degrees to Radians Formulas If x is an angle in degrees and t is an radians then: t angle 1800t t and x — 0) 0) 0) 2 Formulas — cos 0 — sec 0 — cot 0 2 0) — sin 0 0) — CSC 0 0 — tan 0 1800 1800 2
  3. 27T 47T o 60 300 o 3 45 o 1800 2' o 135 1500 2100 225 '2 o 120 240 o Unit Circle 90 2700 o 30 o 330 3150, 00 11 T 6 27T For any ordered pair on the unit circle (x, y) Example cos cos0 — x and sin 0 sm
  4. Inverse Trig Functions 0 — sin— 0 — tan 0 — sin 0 — tan Definition Inverse Properties These properties hold for x in the domain and 0 in the range 0 — cos (x) is equivalent to x (x) is equivalent to x (x) is equivalent to x Domain and Range — sin 0 — cos 0 — tan 0 Range 2 2 sin (sin— cos(cos tan(tan Other Notations sin—I (sin(0)) cos(0)) cos tan—I (tan (0)) Function 0 — cos— Domain 1 1 —oo < < oo 2 sin cos— 2 tan — arcsin x — arccos(x) — arctan(x) Law of Sines, a Law of Sines sin a sin" sin Cosines, and Tangents [3 b c Law of Tangents a b c Law of Cosines 2 a 2 c 2 2 — 2bc cos a — 2ac cos {3 — 2ab COS a—b a—c 4 tan — [3) tan + [3) tan H [3 — 7) tan } ((3 + 7) tan — 7) tan 3 (a + 7)
  5. Complex Numbers .4 —1 —1 -a-iv'ä, a 20 (a + bi)(a — bi) a + bi — (c + di) — (IC — bd —I— (ad bC)i a2 + b2 Complex Modulus — a — bi Complex Conjugate '2 DeMoivre's Theorem Let z — r(cos 0 + i sin 0), and let n be a positive integer. Then: — r n (cos no + i sin no). z 6 Example: Let z — 1 — i, find z Solution: First write z in polar form. —1 —1 0 — arg(z) — tan 1 4 f)) (cos f) Polar Form: z + i Sin Applying DeMoivre's Theorem gives . 6 z 6 f) 2 cos 6 • — + isin 6 • f)) — 23 cos 37T + i Sin 2 37T 2 5
  6. Finding the nth roots of a number using DeMoivre's Theorem Example: Find all the complex fourth roots of 4. That is, find all the complex solutions of 4 We are asked to find all complex fourth roots of 4. 4 These are all the solutions (including the complex values) of the equation x For any positive integer n , a nonzero complex number z has exactly n distinct nth roots. More specifically, if z is written in the trigonometric form r(cos 0 + i sin 0), the nth roots of z are given by the following formula. 1 ( * ) cos 0 3600k + i sin 0 3600k n for k = 0, 1, 2, ...,n— 1. n n Remember from the previous example we need to write 4 in trigonometric form by using: b —1 (a)2 + (b)2 0 — arg(z) and — tan a So we have the complex number a + ib — 4 + i0. Therefore a — 4 and b — 0 (4)2 + (0)2 _ 4 and So r 0 —1 0 — arg(z) — tan 4 Finally our trigonometric form is 4 — 4(cos 00 + i sin 00) Using the formula above with n — 4, we can find the fourth roots of 4(cos 00 + i sin 00) 3600 00 3600 For k For k For k For k 1 47 47 47 cos cos cos cos + i Sin 4 4 3600 * 1 00 3600 * 1 + i Sin 4 4 3600 00 3600 + i Sin 4 4 3600 00 3600 (cos(00) + i sin(00)) — (cos(900) + i sin(900)) — (cos(1800) + i sin(1800)) (cos(2700) + i sin(2700)) 4 + i Sin — 4 are: 4 4 Thus all of the complex roots of x 2, 2i, 2 6
  7. Formulas for the Conic Sections Circle StandardForm : (x — + (y Where (h, k) = center and r Ellipse 2 — radius Standard Form for Horizontal Major Axis : 1 2 a Standard Form for Vertical Major Axis : 1 2 a Where (h, k)= center 2a=length of major axis 2b=length of minor axis 2 Foci can be found by using c Where c= foci length 7
  8. More Conic Sections Hyperbola Standard Form for Horizontal Transverse Axis : 2 a 1 b2 Standard Form for Vertical Transverse Axis : 1 2 a Where (h, k)= center a=distance between center and either vertex Foci can be found by using b2 = c 2 2 a Where c is the distance between center and either focus. (b > O) Parabola Vertical axis: y a(x — h) 2 + k Horizontal axis: x — a(y — + h Where (h, k)= vertex a=scaling factor 8
  9. = sin(T) 3? 5? Emample . sin — cos(T) 3? 5? 7? 7? 5? 5? 4? 3? 4? 3? 5? 5? 1 ?? 1 ?? 2? 2? Emample . cos
  10. 57T 27? — tan ? 10 27? 37? 57?

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