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TRIGONOMETRIC IDENTITIES • Reciprocal identities 1 1 sin u = cos u = csc u sec u 1 1 tan u = cot u = cot u tan u 1 1 csc u = sec u = sin u cos u • Pythagorean Identities sin2 u + cos2 u = 1 1 + tan2 u = sec2 u 1 + cot2 u = csc2 u • Quotient Identities sin u cos u tan u = cot u = cos u sin u • Co-Function Identities π π sin( − u) = cos u cos( − u) = sin u 2 2 tan( csc( π π − u) = cot u cot( − u) = tan u 2 2 sec( π − u) = csc u 2 • Sum-to-Product Formulas sin u + sin v = 2 sin u+v 2 u+v 2 u+v 2 u+v 2 cos u−v 2 u−v 2 u−v 2 u−v 2 • Power-Reducing/Half Angle Formulas sin2 u = 1 − cos(2u) 2 1 + cos(2u) cos2 u = 2 1 − cos(2u) tan2 u = 1 + cos(2u)

sin u − sin v = 2 cos

sin

cos u + cos v = 2 cos

cos

cos u − cos v = −2 sin

sin

π − u) = sec u 2

• Product-to-Sum Formulas sin u sin v = cos u cos v = sin u cos v = cos u sin v = 1 [cos(u − v) − cos(u + v)] 2 1 [cos(u − v) + cos(u + v)] 2 1 [sin(u + v) + sin(u − v)] 2 1 [sin(u + v) − sin(u − v)] 2

• Parity Identities (Even & Odd) sin(−u) = − sin u cos(−u) = cos u tan(−u) = − tan u cot(−u) = − cot u csc(−u) = − csc u sec(−u) = sec u • Sum & Diﬀerence Formulas sin(u ± v) = sin u cos v ± cos 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 • Double Angle Formulas sin(2u) = 2 sin u cos u cos(2u) = cos2 u − sin2 u = 2 cos2 u − 1 = 1 − 2 sin2 u 2 tan u tan(2u) = 1 − tan2 u

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...MAC1114 — College Trigonometry — Project 1 Instructions: Either complete the project on separate paper or type your answers using MS Word. Label each of your problems clearly and in numerical order. Once your project is complete, save the document as a pdf ﬁle and upload your ﬁle to the Dropbox called Project 1 in Falcon Online at class.daytonastate.edu. If you are submitting a handwritten document, you must write NEATLY. If you are submitting a document using MS Word, you must use the Equation Editor correctly. Points will be deducted for work that is not neatly written or the use of incorrect symbols/notation. You need to show all of your work. Part I – Proofs Recall the following deﬁnitions from algebra regarding even and odd functions: • A function f (x) is even if f (−x) = f (x), for each x in the domain of f . • A function f (x) is odd if f (−x) = −f (x), for each x in the domain of f . Also, keep in mind for future reference that the graph of an even function is symmetric about the y-axis and the graph of an odd function is symmetric about the origin. The following shows that the given algebraic function f is an even function. In Project 2 you will need to show whether the basic trigonometric functions are even or odd. Statement: Show that f (x) = 3x4 − 2x2 + 5 is an even function. Proof: If x is any real number, then f (−x) = 3(−x)4 − 2(−x)2 + 5 = 3x4 − 2x2 + 5 = f (x) and hence f is even. Now you should prove the following in a similar......

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