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This guide provides a comprehensive overview of the six trigonometric functions as defined by angles on the unit circle, including specific values for key angles like 30°, 45°, 60°, 90°, and beyond. It covers fundamental identities, such as reciprocal identities, cofunction identities, and Pythagorean identities, along with power-reducing and angle-sum formulas. Through visual representation and algebraic formulas, this resource aims to enhance understanding of trigonometric concepts while facilitating problem-solving skills in trigonometry.
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Definition of the Six Trigonometric Functions Unit Circle q 90o 120o 60o 135o 2p 11p 7p 5p 3p 5p 7p 5p 3p 4p 45o 3 3 2 4 6 6 4 3 4 6 150o 30o p 180o 0 0o y x r y r x 2p 360o y r x x y r 210o 330o 225o 315o 240o 300o 270o 2 3 3 2 2 3 2 2 2 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 p p p p 2 3 4 6 Power-Reducing Formulas 1 – cos 2u 1 + cos 2u sin2 u = cos2 u = 2 2 1 – cos 2u tan2 u = 1 + cos 2u Sum-to-Product Formulas sin u – sin v = 2 cos ( ) sin () sin u + sin v = 2 sin ( ) cos () u + v u – v u + v u – v 2 2 2 2 cos u + cos v = 2 cos ( ) cos () u + v u – v 2 2 p p p p p p – u – u – u – u – u – u cos u – cos v = – 2 sin ( ) sin () u + v u – v 2 2 2 2 2 2 2 2 Product-to-Sum Formulas sin u sin v = ½ [cos( u – v ) – cos( u + v )] cos u cos v = ½ [cos( u – v ) + cos( u + v )] sin u cos v = ½ [sin( u + v ) + sin( u – v )] cos u sin v = ½ [cos( u + v ) – sin( u – v )] sin( ) = cos( ) = u + 1 – cos u 2 – 2 u + 1 + cos u 2 – 2 tan ( ) = = u 1 – cos u sin u 2 sin u 1 + cos u Right triangle definitions, where 0 < q < p/2 (x,y) (cos u, sin u) (0,1) y opp.hyp. sin q =csc q = cos q =sec q = tan q = cot q = hyp. opp. (– , ) ( , ) Hypotenuse Opposite (– , ) ( , ) adj.hyp. hyp. adj. (– , ) ( , ) Adjacent opp.adj. adj. opp. (–1,0) (1,0) x Circular function definitions, where q is any angle. y r = x2 + y2 (– , – ) ( , – ) (x,y) sin q =csc q = cos q =sec q = tan q = cot q = r ( , – ) q (– , – ) y x x (– , – ) ( , – ) (0, –1) Reciprocal Identities Double Angle Formulas 1 1 1 sin u = cos u = tan u = csc u = sec u = cot u = sin 2u = 2 sin u cos u cos 2u = cos2 u – sin2 u = 2 cos2 u – 1 = 1 – 2 sin2 u csc u sec u cot u sin u cos u tan u 1 1 1 2 tan u tan 2u = 1 – tan2 u Quotient Identities Half-Angle Formulas sin u cos u tan u = cot u = cos u sin u Pythagorean Identities sin2 u + cos2 u = 1 The signs of sin(u/2) and cos(u/2) depend on the quadrant in which u/2 lies 1 + tan2 u = sec2 u 1 + cot2 u = csc2 u Cofunction Identities sin() = cos u cos() = sin u tan() = cot u cot() = tan u sec() = csc u csc() = sec u Even/Odd Identities sin(– u) = – sin u cot(– u) = – cot u cos(– u) = cos u sec(– u) = sec u tan (– u) = – tan u csc(– u) = – csc u Sum and Difference Formulas sin(u + v) = sin u cos v + cos u sin v sin(u – v) = sin u cos v – cos u sin v cos(u + v) = cos u cos v – sin u sin v cos(u – v) = cos u cos v + sin u sin v tan u + tan v tan(u + v) = 1 – tan u tan v tan u – tan v tan(u – v) = 1 – tan u tan v RVCC – ASC : RME 4-16-2007
