BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y.

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1 BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH06 Review Sheet. Perform the indicated operations and simplif: n n 0 n + n ( 9 )( ) ab a + b a 6a a a (f) (g) n n 6 n +n (h) (i) 6( ) ( ). Solve: = 6 = +7 n + n = + = 8. Simplif (epress results with positive eponents onl and rationalized denominators): 6 / ( ) (f) 8 z 7 (g) ( 8 6 ) /. Perform the indicated operations and simplif: 0 + ( ) ( ) 6 8 ( ) 6 ( 6+ )( ) 8 7 (f) 7 + 6

2 . Solve for and check our solutions: = += + += + = 6. Solve b factoring: 7 +=0 Solve with the quadratic formula: + +=0 Solve b completing the square: 8 =0 Solve b an method: +0 +6=0 7. Simplif (epress our results in the form a + bi for a, b real): ( 8i) (8 i) (0 + i)( i) ( )( 6 ) i i 7i ( 9+i) (f) (cos 0 + i sin 0 ) 8. Given f() =, determine: f() f( ) f(/) f 9. Sketch the graphs of each of the given functions, indicating the and intercepts, the verte, the ais of smmetr and stating the maimum or minimum value of the function: f() = g() = + h() = ( +) k() = ( +) + w() = Given f() = 9 + Determine the values of for which the function is defined Evaluate: Evaluate: f(0) f( ). Given the functions f() = + and g() = sketch both graphs on the same set of aes.. Sketch each pair of functions on the same set of aes: f() = and g() = f() = and g() = log. Solve for (use the definitions and properties of eponents and logarithms): =8 7 =9 = log = log 6 (6 8 )= (f) log = Page

3 . Find angle θ in the second quadrant if sin θ =/. Find angle θ if tan θ = andcosθ>0. Find two angles θ for which cos θ = /.. Find sin θ if cos θ = / andθ is in Quadrant III. Find cot θ if tan θ =/. Find sec θ if sin θ = / andcosθ<0. 6. Evaluate sin θ, cosθ and tan θ eactl for each of the following angles: θ =0 θ = 0 θ =67 7. For each of the following angles θ, draw them in standard position, choose a specific point on the terminal side of θ and determine the eact values of sin θ, cosθ and tan θ without using a calculator: θ = 6 π θ = θ =70 8. If each of the following points P are on the terminal side of angle θ in standard position with 0 θ<60, draw θ and determine the value of the si trigonometric functions of θ: P =(, ) P =(, ) P =(, ) 9. Solve the following (clearl specif the unknown, draw a labeled diagram if appropriate and state the solution in words): The time a person takes to paddle a kaak miles downstream is the same as the time to paddle half a mile upstream. If the rate of the current is mph, what is the person s paddling rate in still water? Bill is standing on top of a 7 foot cliff overlooking a lake. The measure of the angle of depression to a boat is 0. How far, eactl, is the boat from the bottom of the cliff? Suppose that the height in meters of a golf ball, hit from a tee, is approimated b = t +0t where t is the time in seconds. Find the maimum height of the ball and the time it reaches this maimum height. 0. Graph each equation for π π: = sin =cos = cos. A central angle of θ =60 is contained in a circle with radius r = 0 inches. Find (leaving all results in terms of π): the length of the arc subtended b θ the area of the sector determined b θ 60 0 in Page

4 . Find the eact values of the area and perimeter of this right triangle: B A 0 90 a = in C. Verif the following trigonometric identities: csc θ tan θ cos θ = csc θ sin θ =cotθ cos θ tan θ +=sec θ cos θ sin θ = tan θ +tan θ. A hot-air balloon rises verticall. An observer stands on level ground at a distance of feet from a point on the ground directl below the passenger s compartment. How far, to the nearest foot, does the balloon rise if the angle of elevation changes from 0 to? A state trooper is hiding 0 feet from a straight highwa with a speed limit of 6 mph. One second after a truck passes, the angle θ between the highwa and the line of observation from the patrol car to the truck is measured. (i) If θ = does the truck driver get a speeding ticket ( mile =,80 ft)? (ii) If θ =0 does the truck driver get a speeding ticket? Answers. n n + ( +) + + 9ab +8b a + b. (f) (g) n + (n 9)(n +)(n ) (h) (i) +6 ( ) = = 7 n =/7, 7/ = (f) z z (g) (f) Page

5 = = = ( is an etraneous solution) = 0 (8 is an etraneous solution) =, = ± i =± i 8 6i 8 i = ± i 7 i (f) +i f() = 0 f( ) = f(/) = / = f =a 9. Ais of smmetr =0 Ais of smmetr =0 Verte (0, ) Verte (0, ) Minimum = -intercept (0, ) Maimum = -intercept (0, ) -intercepts (, 0), (, 0) -intercepts (, 0), (, 0) Ais of smmetr = Ais of smmetr = Verte (, 0) Verte (, ) Maimum =0 -intercept (0, ) Maimum = -intercept (0, ) -intercept (, 0) -intercepts ( ±, 0) Ais of smmetr = / Verte ( /, /) Minimum =/ -intercept (0, ) -intercepts none 0. (, ) (, ) (, ) or all real numbers ecept and f(0) = / f( ) = 0 Page

6 . f() g(). f() f() g() g(). = = =/ =0 = 8 (f) =/6.. θ =0 θ = θ =0 and 0 sin θ = / cotθ = secθ = / = / 6. sin 0 = sin 0 = /, cos 0 = cos 0 = /, tan 0 =tan0 = / sin( 0 ) = sin 60 = /, cos( 0 )= cos 60 = /, tan( 0 )= tan 60 = sin 67 = sin = /, cos 67 =cos = /, tan 67 = tan = Page 6

7 7. (, ) (, ) (, ) sin θ = = cos θ = = tan θ = = csc θ = sin θ = cos θ = tan θ = = csc θ = sin θ = = cos θ = = tan θ = = csc θ = sec θ = sec θ = sec θ = cot θ = cot θ = cot θ = Page 7

8 8. (, ) (, ) (0, ) sin 6 π = sin = = sin 70 = = cos 6 π = cos = = cos 70 = 0 =0 tan 6 π = = tan = = tan70 = 0, undefined 9. The person s paddling rate in still water is mph. The boat is 7 feet from the bottom of the cliff. The maimum height of the ball is meters which it reaches in second. 0. = sin π π π π =cos π π π π Page 8

9 = cos π π π π... arc length = 0π inches Area = tan 0 in = in ; perimeter = + in To prove these identities, use algebra and the basic identities area = 0π square inches. The balloon rises feet. tan θ = sin θ cos θ cos θ + sin θ = csc θ = sin θ sec θ = cos θ cot θ = tan θ (i) The truck is traveling at.96 ft/sec which is 76. mph and the driver gets a ticket. (ii) The truck is traveling at.96 ft/sec which is. mph and the driver does not get a speeding ticket. Revised Fall 006 PR, /00, 6/06, /07 C.O S Page 9