Math 151 Appendix D Trigonometry Review Measurement of Angles Angles can be measured in degrees or radians. One complete revolution corresponds to 360°, or 2π radians. So, 360° = 2π radians which gives the conversion formula: π radians = 180° Example: A. Convert 72° to radians. B. Convert !3!4 to degrees. 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. Pythagorean Theorem For any right triangle, with hypotenuse c and sides a and b, c2 = a2 + b2. sin( A) =ac=length of opposite sidelength of hypotenuse=opphypcos( A) =bc=length of adjacent sidelength of hypotenuse=adjhyptan( A) =ab=length of opposite sidelength of adjacent side=oppadj csc( A) =ca=length of hypotenuselength of opposite side=hypoppsec( A) =cb=length of hypotenuselength of adjacent side=hypadjcot( A) =ba=length of adjacent sidelength of opposite side=adjoppMath 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 3 21 23 1 30° 30° 60° 60° 1 2 3 21 23 1 30° 30° 60° 60° 45-45-90 (Isosceles Right) Triangle 22 1 2 1 22 1 45° 45° 45° 45° The Unit Circle 3 2Math 151 Example: Find all trig ratios for ! =2"3. Example: If cos x =15 and 0 ! x !!2, find the values of the other trig functions evaluated at x. Trigonometric Identities In addition to the values on the Unit Circle, you will be expected to readily recall the following identities: The Pythagorean identities: sin2! + cos2! = 1 1+ tan2! = sec2! 1+ cot2! = sin2! The quotient and reciprocal identities: sec! =1cos! csc! =1sin ! tan!=sin!cos! cot ! =cos!sin !=1tan ! The double-angle formulas: sin 2!( )= 2sin! cos!cos 2!( )= 2cos2! !1 " cos2! =121+ cos 2!( )( )cos 2!( )= 1! 2 sin2! " sin2! =121! cos 2!( )( ) Example: Solve the following equations for x, where 0 ! x ! 2!. A. 2cos x !1 = 0 B. 2cos x + sin 2x = 0Math 151 Example: If sec x =53 and !!2< x < 0, what is the value of sin 2x? 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
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