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 )( ) + + 6 + 9ab a+b a 6a a a (f) + 0 + (g) n n 6 n +n (h) (i) 6( ) ( ). Solve: 7 + 9 = 6 = +7 n+ n = + = 8. Simplif (epress results with positive eponents onl and rationalized denominators): 6 / ( ) 8 7 6 (f) 8 z 7 (g) ( 8 6 ) /. Perform the indicated operations and simplif: 0 + ( ) ( ) 6 8 ( ) 6 ( 6+ )( ) 8 7 (f) 7 + 6
. 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) ( )( 6 ) 7i( 9+i) (0+i)( i) i i (f) (cos0 +isin0 ) 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() = ++ 0. Given f() = 9 + Determine the values of for which the function is defined Evaluate: f(0) Evaluate: f( ). Graph: + 6 <. 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
. Solve and write the answer in set notation: = = > 6. Find the center and radius for each circle: + = + = 9 6+ = 8 7. Find θ, to the nearest whole degree, if sinθ = 0. and cosθ < 0 Find sinθ if cosθ = / and θ is in Quadrant III. 8. Without using a calculator, determine the eact value of: sin (0 )+cos (0 ) tan(60 ) sec ( ) tan ( ) sin(π/6) cos(π/) 9. IfeachofthefollowingpointsP areontheterminalsideofangleθ instandardpositionwith0 θ < 60, draw θ and determine the value of the si trigonometric functions of θ: P = (, ) P = (,) P = (,) 0. 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 sinθ, cosθ and tanθ without using a calculator: θ = 6 π θ = θ = 70. Evaluate sinθ, cosθ and tanθ eactl for each of the following angles: θ = 0 θ = 0 θ = 67. 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 9. How far is the boat from the bottom of the cliff (rounded to one decimal place)? 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.. Graph each equation for π π: = sin = cos = cos. A central angle of θ = 0 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 0 in 0 Page
. Find the eact values of the area and perimeter of this right triangle: B A 0 90 b = in C 6. Verif the following trigonometric identities: cscθtanθcosθ = cscθ sinθ = cotθcosθ tan θ + = sec θ cos θ sin θ = tan θ +tan θ 7. 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 0? 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/ =.. 8 6 0 8 6 (f) z z (g) 8 0+ 8 6 +6 0 8 + 7 7 (f) Page
. 6. 7. 8. = = = ( 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 Verte (0, ) Minimum = -intercept (0, ) -intercepts (,0), (,0) Ais of smmetr = 0 Verte (0, ) Maimum = -intercept (0, ) -intercepts (, 0), (, 0) Ais of smmetr = Verte (, 0) Maimum = 0 -intercept (0, ) -intercept (, 0) Ais of smmetr = Verte (, ) Maimum = -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
.. f() g(). f() g() f() g(). = = = / = 0 = 8 (f) = /6 Page 6
. {,} {,} { } { < or > } { / or } 6. center (0,) radius center (0,0) radius center (,) radius 7. θ = 7 sinθ = / 8. / / = ( )/ 9. (, ) (,) (, ) sinθ = = sinθ = sinθ = = cosθ = = tanθ = = cscθ = secθ = cosθ = tanθ = = cscθ = secθ = cosθ = = tanθ = = cscθ = secθ = cotθ = cotθ = cotθ = Page 7
0. (,) (, ) (0, ) sin 6 π = cos 6 π = sin = = cos = = sin70 = = cos70 = 0 = 0 tan 6 π = = tan = = tan70 = 0, undefined. sin0 = sin0 = /, cos0 = cos0 = /, tan0 = tan0 = / sin( 0 ) = sin60 = /, cos( 0 ) = cos60 = /, tan( 0 ) = tan60 = sin67 = sin = /, cos67 = cos = /, tan67 = tan =. 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 0 meters which it reaches in second.. = sin π π π π Page 8
= cos π π π π = cos π π π π.. 6. Area = tan0 in = arc length = π inches in ; perimeter = + in To prove these identities, use algebra and the basic identities area = 7π square inches 7. The balloon rises 7 feet. tanθ = sinθ cosθ cos θ +sin θ = cscθ = sinθ secθ = cosθ cotθ = tanθ (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. Spring 00 MB, Spring 00, Fall 006 PR. Revised /00, C.O S. Page 9