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Trigonometry Second Semester Final Study Sheet *to convert degrees to radians; multiply the degree by rad * to convert radians to degrees; multiply the radians by 180 180 rad Coterminal angles: two angles with the same initial sides but possibly different rotations. - every angle has infinitely many coterminal angles - is coterminal with angles is coterminal with angles 360 k (k is an integer) 2 k *Length of a Circular Arc s r ( in radians) Definitions of the Trig functions in Terms of a Unit Circle sin t = y csc t = 1 cos t = x sec t = 1 x y x cot t = x y tan t = Even Functions cos(-t) = cos t sec(-t) = sec t y Odd Functions sin(-t) = -sin t csc(-t) = -csc t tan(-t) = -tan t cot(-t) = -cot t Reciprocal Identities 1 csc t 1 cos t sec t 1 tan t cot t sin t 1 sin t 1 sec t cos t 1 cot t tan t csc t Quotient Identities tan t sin t cos t cot t cos t sin t Pythagorean Identities sin 2 t cos 2 t 1 1 tan 2 t sec 2 t 1 cot 2 t csc 2 t SOHCAHTOA Cofunction Identities The value of a trigonometric function of is equal to the cofunction of the complement of Cofunctions of complementary angles are equal. sin cos 90 cos sin 90 tan cot 90 cot tan 90 sec csc 90 If is in radians, replace 90 with csc sec 90 2 . Definitions of Trigonometric Functions of Any Angle Let . be any angle in standard position and let P = (x, y) be a point on the terminal side of . If r x 2 y 2 is the distance from (0,0) to (x,y), as shown in the figure above, the six trigonometric functions of are defined by the following ratios: sin y r cos x r tan y ,x 0 x r ,y0 y r sec , x 0 x x cot , y 0 y csc Sum and Difference Formulas for Cosines and Sines cos cos cos sin sin cos cos cos sin sin sin sin cos cos sin sin sin cos cos sin Sum and Difference Formulas for Tangents tan tan tan 1 tan tan tan tan tan 1 tan tan Three forms of the Double-Angle Formula of cos 2 Double Angle Formulas sin 2 2sin cos cos 2 cos 2 sin 2 cos 2 cos 2 sin 2 cos 2 2 cos 2 1 tan 2 cos 2 1 2sin 2 2 tan 1 tan 2 Power Reducing Formulas 1 cos 2 2 1 cos 2 cos 2 2 1 cos 2 tan 2 1 cos 2 sin 2 Half-Angle Formulas 1 cos 2 2 1 cos cos 2 2 1 cos tan 2 1 cos sin Half-Angle Formulas for Tangent tan 2 1 cos sin tan 2 sin 1 cos Product-to-Sum Formulas 1 cos cos 2 1 cos cos cos cos 2 1 sin cos sin sin 2 1 cos sin sin sin 2 sin sin Sum-to-product Formulas sin sin 2sin cos 2 2 sin sin 2sin cos 2 2 cos cos 2 cos cos 2 2 cos cos 2sin sin 2 2 Sum-to-product Formulas sin sin 2sin cos 2 2 sin sin 2sin cos 2 2 cos cos 2 cos cos 2 2 cos cos 2sin sin 2 2 The Law of Sines If A,B and C are the measures of the angles of a triangle, and a, b, and c are the lengths of the sides opposite these angles, then a b c sin A sin B sin C The Law of Cosines If A, B and C are the measures of the angles of a triangle, and a, b and c are the lengths of the sides opposite these angles, then a 2 b 2 c 2 2bc cos A b 2 a 2 c 2 2ac cos B c 2 a 2 b 2 2ab cos C The Sign of r and a Point’s Location in Polar Coordinates The point P = (r, ) is located r units from the pole. If r>0, the point lies on the terminal side of . If r<0, the point lies along the ray opposite the terminal side of r = 0, the point lies at the pole, regardless of the value of . . If Relations between Polar and Rectangular Coordinates x r cos y r sin x2 y 2 r 2 y tan x Converting a Ponit from Rectangular to Polar Coordinates ( r 1) Plot the point (x, y). 0 and 0 2 ) 2) Find r by computing the distance from the origin to (x, y): 3) Find using tan y x with the terminal side of The Absolute Value of a Complex Number The absolute value of the complex number a + bi is z a bi a2 b2 r x2 y 2 passing through (x, y). Polar Form of a Complex Number The complex number z = a + bi is written in polar form as z r (cos i sin ) , where a r cos , b r sin , r a2 b2 and tan b . The value of r is called the a modulus (plural: moduli) of the complex number z and the angle complex number z with 0 2 . is called the argument of the Product of Two Complex Numbers in Polar Form If z1 r1 (cos 1 i sin 1 ) and z2 r2 (cos 2 i sin 2 ) z1 z2 r1r2 [cos(1 2 ) i sin(1 2 )] Quotient of Two Complex Numbers in Polar Form If z1 r1 (cos 1 i sin 1 ) and z2 r2 (cos2 i sin 2 ) z1 r1 cos 1 2 i sin 1 2 z2 r2 DeMoivre’s Theorem Let z r cos the nth power, is i sin be a complex number in polar form. If n is a positive integer, then z to z n r cos i sin r n cos n i sin n n DeMoivre’s Theorem for Finding Complex Roots Let w r cos i sin be a complex number in polar form. If complex nth roots given by the formula 2 k 2 k zk n r cos (radians) i sin n n 360 k 360 k n zk r cos n i sin n The magnitude of v = ai + bj is given by v a 2 b2 (degrees) w 0 , w has n distinct Representing Vectors in Rectangular Coordinates Vector v with initial point vector P1 ( x1 , y1 ) and terminal point P2 ( x2 , y2 ) v x2 x1 i y2 y1 j is equal to the position