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
| Sine and Cosine Laws Normally to use the sine and cosine functions, we used right triangles. Now, we can use the laws of cosine and sine to solve any type of triangle. Law of Sines:
Proof: Sin A = h/b also written as bSinA = h Sin B = h/a also written as aSinB = h Using Substitution, we conclude that bSinA = aSinB Divide both sides by ab SinA/a = SinB/b By drawing an altitude from A, and following the same procedure, we can conclude that SinB/b = SinC/c SinA/a = SinB/b = SinC/c The Law of Sines is useful to solve a triangle when the only given information is one angle and two sides where the angle is between the two sides, or two angles and one side. Law of Cosines: a2 = b2 + c2 - 2bcCosA b2 = a2 + c2 - 2acCosB c2 = a2 + b2 - 2abCosC In words, the Law of Cosines means that in any triangle, the square of the length of one side is equal to to the sum of the squares of the lengths of the other two sides subtracting two times the product of the lengths of these sides and the cosine of the included angle. It is useful to use the law of cosines when you are given a triangle with three known sides, or one angle and two sides, where the angle is not between the two given sides. To find and angle in the triangle when given three sides, it is easier to switch around the equations and use: CosA = (b2 + c2 - a2)/2bc CosB = (a2 + c2 - b2)/2ac CosC = (a2 + b2 - c2)/2ab
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