Math 151 Appendix D Trigonometry Review Measurement of Angles Angles can be measured in degrees or radians One complete revolution corresponds to 360 or 2 radians So 360 2 radians which gives the conversion formula radians 180 Example A Convert 72 to radians B Convert 3 to degrees 4 Trigonometric Ratios The hypotenuse of a right triangle is the longest side and is opposite the right angle Trigonometric functions are defined as ratios of the lengths of the sides of a right triangle These are the basis for the trigonometric functions in which we replace the triangle by points on a plane a length of opposite side opp sin A c length of hypotenuse hyp b length of adjacent side adj c length of hypotenuse hyp a length of opposite side opp tan A b length of adjacent side adj cos A c length of hypotenuse hyp a length of opposite side opp c length of hypotenuse hyp sec A b length of adjacent side adj b length of adjacent side adj cot A a length of opposite side opp csc A Pythagorean Theorem For any right triangle with hypotenuse c and sides a and b c2 a2 b2 Math 151 30 60 90 Triangle In a 30 60 90 triangle the length of the side opposite the 30 angle is half the length of the hypotenuse 1 2 1 60 2 1 60 1 1 60 60 1 2 2 30 30 30 30 3 2 3 2 3 45 45 90 Isosceles Right Triangle 2 2 45 45 1 2 1 2 45 2 45 1 The Unit Circle 3 2 3 Math 151 Example Find all trig ratios for Example If cos x 2 3 1 and 0 x find the values of the other trig functions evaluated at x 5 2 Trigonometric Identities In addition to the values on the Unit Circle you will be expected to readily recall the following identities The Pythagorean identities sin 2 cos 2 1 1 tan 2 sec 2 1 cot 2 sin 2 The quotient and reciprocal identities sec 1 cos csc 1 sin tan sin cos The double angle formulas sin 2 2sin cos cos 2 2cos 2 1 cos 2 1 2sin 2 1 1 cos 2 2 1 sin 2 1 cos 2 2 cos 2 Example Solve the following equations for x where 0 x 2 A 2cos x 1 0 B 2cos x sin 2x 0 cot cos 1 sin tan Math 151 Example If sec x 5 and x 0 what is the value of sin 2x 3 2 Graphs of Sine Cosine and Tangent Functions Example Sketch a graph of f x 1 cos x Example Sketch a graph of f x tan x 2
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