ࡱ>  rY| @ bjbjFF "`,,2222222Fn<FsM2LLLLLLL$NRPfM2M22-M---22L-L--:J22L r%KLCM0sM K]Q'^]Q4LFF2222]Q2LP-MMFFd+FFTRIGONOMETRY REVIEW Circles Angles measured using degrees, radians, or revolutions One full revolution around a circle is equal to 360( and 2( radians Degree or radian measurements are equivalent over full revolutions E.g. 90( is equivalent to 450(, 90( plus 360(k where k is an integer Arc length Arc length for whole circle (or circumference) is 2(r Arc length is equal to (r Example: Arc Length of circle of radius 2 at (=(/2  arc length = (/2 (2) =( Area of a sector: area of whole circle is (r2 the area of a sector with angle ( is  EMBED Equation.3  Trigonometric Identities  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The phrase SOH CAH TOA is a very useful when finding sin (, cos (, tan ( when given the lengths of sides of a triangle. Sin=Opposite/Hypotenuse Cos=Adjacent/Hypotenuse Tan=Opposite/Adjacent  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Proofs sin(A+B)   EMBED Equation.3  cos(A+B)   EMBED Equation.3 Sorry Arup I was having trouble understanding what to type for the proof. tan(A+B)  EMBED Equation.3  sin2(  EMBED Equation.3  cos2 EMBED Equation.3   EMBED Equation.3  tan2 EMBED Equation.3   EMBED Equation.3  Finding Ratios with out finding the angle. When given a ratio, for example  EMBED Equation.3  it is possible to find other ratios such as cos EMBED Equation.3 , tan EMBED Equation.3 , sin2 EMBED Equation.3 , cos2 EMBED Equation.3  EMBED Equation.3 , etc. To find the other trig functions with a single  EMBED Equation.3 , just use the Pythagorean theorem. Using the example  EMBED Equation.3 :  52-22=x2 or 25-4=21=x2 So cos EMBED Equation.3 =  EMBED Equation.3 , tan EMBED Equation.3 =  EMBED Equation.3 , etc. To find trig functions with more than 1  EMBED Equation.3 , manipulate the trigonometric identities. E.g. give that  EMBED Equation.3  to sin2 EMBED Equation.3  use sin2 EMBED Equation.3 =2sin EMBED Equation.3 cos EMBED Equation.3  2sin EMBED Equation.3 cos EMBED Equation.3 = EMBED Equation.3  The Nature of trigonometric graphs Sinx Has period of 2( Is always continuous Cosx Has period of 2( Is always continuous Similar to graph of sinx but offset by (/2 to the left Tanx Has period of ( Is undefined where x=(/2 + k( K is an integer Composite Function f(x) = a sin(b(x+c)) + d a changes the amplitude of the graph. b changes the period of the graph, condensing or expanding it c moves the graph left or right d moves the graph up or down. Inverse Functions arcsin x: has a range of (- EMBED Equation.DSMT4 , EMBED Equation.DSMT4 )  has a domain of [-1,1] arccos x: has a range of (- EMBED Equation.DSMT4 , EMBED Equation.DSMT4 )  has a domain of [-1,1] arctan x: has a range of [- EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ]  has a domain of (- EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ) Rules and Formulas for Triangles A =  EMBED Equation.DSMT4 bh Heros Formula: A =  EMBED Equation.DSMT4  where s =  EMBED Equation.DSMT4  Cosine Rule:  EMBED Equation.DSMT4  The Sine Rule:  EMBED Equation.DSMT4  Alternate Area: A =  EMBED Equation.DSMT4 absinC Half Angle Identities: We know that  EMBED Equation.DSMT4 . From this it can be determined that  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . If we replace 2A by  EMBED Equation.DSMT4  and use t to denote  EMBED Equation.DSMT4  we have:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  More Identities  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Practice Problems 1) A chord AB divides a circle of radius 2m into two segments. If AB subtends an angle of  EMBED Equation.DSMT4  at the center of the circle, find the area of the minor segment. 2) Find the sine, cosine, and tangent of  EMBED Equation.DSMT4 . 3) Find the general solution of  EMBED Equation.DSMT4 . 4) Solve  EMBED Equation.DSMT4  for angles in the range  EMBED Equation.DSMT4 . 5) Without using tables or a calculator, evaluate: (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c) EMBED Equation.DSMT4  6) Prove that:  EMBED Equation.DSMT4  7) Express  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4 . Solutions 1.   EMBED Equation.DSMT4  Area of sector AOB =  EMBED Equation.DSMT4  Area of  EMBED Equation.DSMT4 AOB =  EMBED Equation.DSMT4  Area of Minor Segment (shaded in) = (2.094 1.732)  EMBED Equation.DSMT4  = .362 EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  because the tangent function has a period of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5. (a) EMBED Equation.DSMT4  (b) EMBED Equation.DSMT4  (c) EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. Using  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  gives  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Hence  EMBED Equation.DSMT4   EMBED Equation.DSMT4  ST   * + ^ _ ` a x y z {     / 0 1 2 3 L jh#XhxEHUj~1B hxCJUVaJ hxH* h#XhxjhxU jqhx jphx jhxh#Xhx>* hx>*hx hx5>T + a { 3 L d | $  & Fgdx & Fgdx & Fgdxgdx$a$gdxL M ` a b c d e x y z { | } µ}naj_h#XhxEHUjB1B hxCJUVaJjSh#XhxEHUj,1B hxCJUVaJjh#XhxEHUj1B hxCJUVaJjh#XhxEHUjh1B hxCJUVaJj1h#XhxEHUj41B hxCJUVaJ h#Xhxjh#XhxU  ! 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