TRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)

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1 TRIG REVIEW NOTES Convert from radians to degrees: multiply by Convert from degrees to radians: multiply by Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal) To find co-terminal angles: add or subtract revolutions of a circle. 4 6 Definitions of Sine, Cosine, and Tangent sin O or y cos A or tan O or y H r H r A r y Arc Length s r s arc length, r radius, angle in radians Reference Angle - an acute angle formed by the terminal side of and the horizontal ais. QUAD I is the reference angle. QUAD II or QUAD III or QUAD IV or Special Triangles and Angles

2 Related Special Angles Graphing y a cos( b c) d y a sin( b c) d Amplitude Amp a Period Per b Horizontal Shift HS set b c 0 and solve for Vertical Shift VS d Key Pts ( period) (0,,,,1) 4 4 y a sec( b c) d y a csc( b c) d Find the information for cosine and sine and plot the 5 key points. Where the key points hit the -ais or VS, put in vertical asymptotes. At the "high" key points, draw a parabolic shape pointing up. At the "low" key points, draw a parabolic shape pointing down. y a tan( b c) d a = since tangent s range (, ), it is not the amplitude but it is the height of some key points. b c We will set this = to the "asymptotes" and solve. This will take care of Per and HS. b c and b c d = This is still the VS. 3 key points 1 st key point = midpoint between asymptotes nd and 3 rd = ½ way between midpoint and asymptote

3 y a cot( b c) d Aysmptotes : b c 0 b c 3 key points: find the same way as in tangent Cosine and Sine are the only trig functions that have an amplitude and have no asymptotes Inverse Trig Functions To have an inverse, the function must be 1 to 1. Since the trig functions are not 1 to 1, we have to choose an interval that is 1 to 1. Domain Range y sin 1 y 1 y cos 0 1 y 1 y tan y So for inverse functions, the domain and the range switch. Domain Range 1 y sin 1 1 y 1 y cos y 1 y tan y This is why arcsin() and arccos(-4) are not possible.

4 We will be asked to evaluate inverse trigonometric epressions. Be careful, sin and sin are NOT the same. The statement on the left has (or more) answers because it is asking what angles have a sine of ½. The statement on the right is asking what the arcsin of ½ is for which there is only 1 answer. (Equation = + answers, inverse epression = 1 answer). 1 1 sin(sin ( )) and cos(cos ( )) only if tan(tan ( )) always sin (sin( )) cos (cos( )) tan (tan( )) and and only if is in their domain If is not in their domain, the statement will an angle that is. Trigonometric Identities Reciprocal: sin cos tan csc sec cot sin cos Quotient: tan = cot cos sin Pythagorean: sin cos 1 1 tan sec 1 cot csc Co-Function: sin u cos u tan u cot u cscu sec u Even & Odd: sin( u) sin( u) cos( u) cos( u) tan( u) tan( u)

5 Sum Difference sin( u v) sin u cos v cosu sin v sin( u v) sin u cos v cosu sin v cos( u v) cosu cos v sin u sin v cos( u v) cosu cos v sin u sin v tan u tan v tan u tan v tan( u v) tan( u v) 1tan u tan v 1tan u tan v Double Angle: sin( ) sin cos cos( ) cos sin u u u u u u cos u 1 1sin u 1cos u 1cos u Power Reducing: sin u cos u Half Angle Formulas u 1 cos 1 cos 1 cos sin sin cos tan u u u u u u sin u 1 cosu u (The depends on which quadrant is in. It will be + or -, never both) Trigonometry Assignment to follow.

6 TRIG SUMMER HOMEWORK WS *Do work on another sheet of paper. DO NOT use a calculator!* Label sides and angles of the special triangles. Which trig functions are positive in: QI QII QIII QIV Find sine, cosine, and tangent for the given angles without a calculator! Find the angles that satisfy the equation without a calculator! Give the answers in degrees and radians csc 8. tan 1 9. cos Graph the following without a calculator 10. y 3cos( ) y sin 1. y tan y sec y csc 15. y cot Simplify using trig identities. 16. tan( ) 17. cot cos cot 0. 1 sin 1. sec tan

7 . cos( y) 3. sin cos y cos sin y 4. 4sin 3 cos 3 5. sin tan cos y y sin cos 6. tan cos sin cos 1 sin cos 3 3 y y y Evaluate without a calculator sin 30. cos 31. tan 3 Evaluate without a calculator sin sin 33. arccos cos tan tan sin arcsin Hint: draw a triangle tan cos 37. cos arcsin

8 Limits and Continuity Notes A limit fails to eist if: a ) lim f ( ) lim f ( ) b ) ( nobound ) c ) Oscillates a a To evaluate a limit: If is approaching a number, plug the number into the function. If this value is undefined: Factor cancel Get a LCD Rationalize the numerator If this does not help, use the graphic/numeric method. If is approaching infinity, think horizontal asymptote. Look at the highest power in n and d If n < d limit = 0 If n = d limit = coefficient of n / coefficient of d If n > d limit = + or infinity Polynomials will equal + or infinity. Please state which one and DNE Piecewise functions: if it splits at the number is approaching, you must show the limit from the left and the right of the value. If f() is continuous or has a hole at = c, then lim f( ) = where the y value is or should be c If f() has an asymptote or a break at = c, then lim f( ) DNE Rational functions have asymptotes where the denominator = 0 and the factor does not cancel c Tangent, Cotangent, Cosecant, Secant all have asymptotes Piecewise functions and Absolute Values may have breaks Special Trig Limits sin 1 cos lim 1 lim 0 0 0

9 Continuity Polynomials, sine, cosine, radicals, eponential, and logarithms are continuous everywhere in their domain. Rational functions are continuous ecept where the denominator = 0. A hole occurs where a factor in the denominator cancels with a factor in the numerator. Definition: A function f is said to be continuous at point c if the following conditions are satisfied: 1). f ( c) is defined ). lim f ( ) eist c lim f ( ) lim f ( ) c c 3). lim f ( ) f ( c) c Types of Discontinuity: Removable and Non-removable Removable means that the discontinuity could be fied by reassigning the value of f(c). Usually removable discontinuities are holes in the graph. Asymptotes and gaps in the graph would be non-removable. Intermediate Value Theorem (IVT) If f() is continuous over [a,b] and k is any number between f(a) and f(b). Then there is at least 1 number, c in [a,b] for which f(c)=k. F(b) F(a) a c b Eample: If f(a) is negative and f(b) is positive, there must be a value in between a and b, where the y value is 0. You would use IVT to see if a function has -ints (zeros)

10 Definition of a derivative dy f '( ) y ' d lim 0 f ( ) f ( ) 4 Reasons for Non-differentiability 1. Corner: y. Cusp: y /3 3. Vertical Tangent: y 1/3 4. Discontinuity (holes, breaks, Vertical Asymptotes): y 1 **Differentiability implies Continuity** (If the function has a derivative, the function is continuous)

11 EXAMPLES Find the limit. 1, lim. lim f ( ) f ( ) 4, 3 4 3, 3 lim 3 9 lim since lim f ( ) lim f ( ) lim f( ) DNE 3 4 im lim 4. l 0 Try something check highest powered terms lim lim 4 ( 4)( 4) 4 ( 4) 8 n d 3 lim 3

12 State where the function is discontinuous and state as removable or non-removable f ( ) 6. f ( ) e 3 (1)( 3) f( ) continuous, ( 1)( 3) discontinous at 1 Not Rem 3 Rem 7. f ( ) 4 tan 8. 1, 1 f ( ) 1, 1 Tangent has asymptotes Each piece is continuous where it is used Check continuity at 1 1 1) f (1) ) lim 1 1 lim Discontinuous at...,,... lim f ( ) Not Rem 3) f (1) lim f ( ) 1 f( ) is continuous, Limit and Continuity Assignment to follow

13 Limits Summer Assignment Worksheet Find the limit lim. lim lim sin cos , 1 ( ), 4. lim f ( ) f ( ) 5. lim g( ) g( ) 1.5( 1), 1, lim 7. lim 8. lim cos (sin ) lim 10. lim 11. lim What can cause a limit to fail to eist? 13. What must be true about f() if lim f ( ) f ( a)? a

14 Continuity Summer Assignment Worksheet 1. f() is said to be continuous at = c if: a) b) c). When do you have to actually show the definition of continuity? 3. List the types of discontinuity and give an eample of each using a function. a) b) Are the following continuous? Why or why not? State whether the discontinuities are removable or not f ( ) g( ) 4 6. h( ) 7. f ( ) tan 6 1 1, 1, 1 8. g( ) 9. h( ) 1 31, 1 0, Does f ( ) 3 4 have a solution/zero in the interval [,3]? Eplain why.

15 Derivative Review 1 ST Derivative is the slope of the tangent line or the instantaneous rate of change. 1 st dy derivative = mtan,, f '( ), y ' d The Definition of a Derivative f ' m lim f ( ) f ( ) tan 0 The slope of the tangent line is the slope of a secant line where the points are pushed closer together until they are one point. The limit does the pushing. Derivative Theorems d For constant functions: [ c ] 0 d The Power rule: d [ n] n d n1 The Sum and Difference rule: d [ f ( ) g ( )] f '( ) g '( ) d The Product Rule: d f ( ) g( ) f '( ) g( ) g '( ) f ( ) The Quotient Rule: f ( ) f '( ) g( ) g '( ) f ( ) d g ( ) ( g ( ))

16 The trig functions: d d d [sin ] cos [cos ] sin [tan ] sec d d d d [csc ] csc cot d [sec ] sec tan d [cot ] csc d d d Eamples Find the derivative f g 1. ( ) ( ) 3 sin 5 1/ 5 f ( ) g '( ) f ' g g ' f f g 1/ 6 '( ) '( ) 6 (sin ) cos (3 ) 10 f '( ) 6 g '( ) 6sin 3 cos cos 3. g( ) 4. h( ) sin f ' g g ' f sin g'( ) h( ) sin g sin 3 0( 5 1) (6 5) 7 g '( ) h '( ) cos 3 ( 51) 4 35 g'( ) ( 51) 3 Position, Velocity, and Acceleration Position s(t) Where the particle is relative to the origin (and initial point) as a function of time. If s(t) < 0, particle is to the left of the origin. If s(t) > 0, particle is to the right of the origin.

17 Velocity v(t) Rate = distance time Change in distance s s( t) s( t1) Average Velocity Change in time t tt1 Instantaneous Velocity = s (t) Speed = v(t) speed has no direction Acceleration a(t) Rate of change in velocity Change in velocity v v( t) v( t1) Average Acceleration = t t Change in time Instantaneous Acceleration = v (t) = s (t) t 1 A particle is slowing down if the velocity and acceleration have different signs. It is speeding up when they have the same sign (+,+ or -,-)