Trigonometry
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Trigonometry Reference Home Back to Thinkquest Contents
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Definitions of Trigonometric Functions Trigonometric Functions of Sums and Differences Inverse Trigonometric Functions
| Trigonometric Identities
The reader should be aware of the above diagram which was defined in the definitions of trigonometric functions page.
An identity is an equation that is true for all values of the variables for which the two sides of the equation are defined. For instance: (a+b)2 = a2 + 2ab + b2 is true for all values of a and b. Trigonometric identities involve the sin(X), cos(X), tan(X), etc. Using the right triangle OBP in the above diagram, OB = Cos(X), BP = Sin(X), and OP = 1 (the radius of the circle). By Pythagorean Theorem, BP2 + OB2 = OP2 Substituting for OB, BP, and OP: Sin2(X) + Cos2(X) = 1 The following diagram shows the similar triangles OBP and OEC extracted from the above diagram.
In similar triangles, the ratios of the similar sides are equal. CE/OE = PB/OB Substituting : Tan(X)/1 = Sin(X)/Cos(X) Tan(X) = Sin(X)/Cos(X) The following diagram shows similar triangles FOD and AOP extracted from the unit circle.
FD/FO = AP/AO Cot(X)/1 = Cos(X)/Sin(X) Cot(X) = Cos(X)/Sin(X) Combining the two identities above, 1/Tan(X) = 1/{Sin(X)/Cos(X)} = Cos(X)/Sin(X) = Cot(X). Therefore: Cot(X) = 1/Tan(X) Using these identities, we can form new ones such as: Tan2(X) + 1 = Sec2(X) = 1/Cos2(X) where Sec(X) = 1/Cos(X) The proof is as follows: Tan2(X) + 1 = Sin2(X)/Cos2(X) + 1 = Sin2(X)/Cos2(X) + Cos2(X)/Cos2(X) = (Sin2(X) + Cos2(X))/Cos2(X) = (1)/Cos2(X) = Sec2(X) Another identity that can be formed is: Cot2(X) + 1 = Csc2(X) where Csc(X) = 1/Sin(X) With a proof of: Cot2(X) + 1 = Cos2(X)/Sin2(X) + 1 = Cos2(X)/Sin2(X) + Sin2(X)/Sin2(X) = (Cos2(X) +Sin2(X))/Sin2(X) = 1/Sin2(X) = Csc2(X)
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