Math 2412 Pre Calculus TEST 2 Prep Fall 2011

Transcription

1 Math 41 Pre Calculus TEST Prep Fall 011 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the eact value under the given conditions. 1) sin α = 3, 0 < α < π ; cos β = 0 9, 0 < β < π Find tan (α + β). Use the given information given to find the eact value of the trigonometric function. ) sin θ = - 0 9, θ lies in quadrant IV Find cos θ. Find the eact value b using a sum or difference identit. 3) sin (18-6 ) 4) sin + sin cos + cos =? ) sin - cot =? 6) sin cos + cos sin =? 7) cos + cos 3 Dr. James Math 41 PreCalculus 10/4/011

2 Describe the graph using another equation. 8) = cos ( + π ) - cos ( - π ) 3 π -3 9) (sin + cos ) 1 + sin cos =? Epress the product as a sum or difference. 10) sin sin Use a half-angle formula to find the eact value of the epression. 11) cos 3π 8 Solve the equation on the interval [0, π). 1) sin + sin = 0 13) sin ( + π ) =? Use trigonometric identities to find the eact value. 14) tan 0 + tan tan 0 tan 10 Solve the equation on the interval [0, π). 1) sin - sin cos = 0 Rewrite the epression as an equivalent epression that does not contain powers of trigonometric functions greater than 1. 16) 8 sin cos 17) sin 3 - sin Dr. James Math 41 PreCalculus 10/4/011 Page

3 18) 8 sin cos3 + 8 sin3 cos =? Find the eact value of the epression. 19) sin cos 1 - cos sin 1 Find the eact value under the given conditions. 0) cos α = - 7, π < α < π; sin β = - 1, π < β < 3π Find tan (α + β). 1) tan α = 4 7, π < α < 3π ; cos β = - 0 9, π < β < π Find tan (α + β). Find the eact value of the epression. ) cos π 1 sin π 4 - cos π 4 sin π 1 Find the eact value b using a difference identit. 3) tan 7 4) cos (α - β) cos α cos β =? Use the given information to find the eact value of the epression. ) Find cos θ. sin θ = 7, θ lies in quadrant I. 6) sin (α + β) sin (α - β) =? 7) If a projectile is fired at an angle θ and initial velocit v, then the horizontal distance traveled b the projectile is given b D = 1 16 v sin θ cos θ. Epress D as a function of θ. Use the given information given to find the eact value of the trigonometric function. 8) sin θ = 1 4, θ lies in quadrant I Find sin θ. Solve the equation on the interval [0, π). 9) csc 3 = 0 Dr. James Math 41 PreCalculus 10/4/011 Page 3

4 Rewrite the epression as an equivalent epression that does not contain powers of trigonometric functions greater than 1. 30) sin3 Find the specified vector or scalar. 31) u = i - j and v = 8i + 7j; Find v - u. Find another representation, (r, θ), for the point under the given conditions. 3) 9, π 3, r > 0 and -π < θ < 0 The rectangular coordinates of a point are given. Find polar coordinates of the point. 33) (, -) 34) Two forces, F1 and F, of magnitude 60 and 70 pounds, respectivel, act on an object. The direction of F1 is N40 E and the direction of F is N40 W. Find the magnitude and the direction angle of the resultant force. Epress the direction angle to the nearest tenth of a degree. Use a graphing utilit to convert from rectangular to polar coordinates. Epress the answer in radians. 3) (-8, -6) Epress θ in thousandths of radians. Solve the equation in the comple number sstem. 36) 3-1i = 0 Write the comple number in rectangular form. 37) 7(cos 3π 4 + i sin 3π 4 ) Use the dot product to determine whether the vectors are parallel, orthogonal, or neither. 38) v = i, w = -3i Use Heron's formula to find the area of the triangle. Round to the nearest square unit. 39) a = 1 meters, b = 18 meters, c = 7 meters Find the quotient z 1 z 40) z1 = 6(cos 3π z = 1(cos π 6 of the comple numbers. Leave answer in polar form. + i sin 3π ) + i sin π 6 ) Dr. James Math 41 PreCalculus 10/4/011 Page 4

5 41) Two sailboats leave a harbor in the Bahamas at the same time. The first sails at 0 mph in a direction 30. The second sails at 33 mph in a direction 10. Assuming that both boats maintain speed and heading, after 3 hours, how far apart are the boats? The rectangular coordinates of a point are given. Find polar coordinates of the point. 4) (-4, -4 ) Convert the rectangular equation to a polar equation that epresses r in terms of θ. 43) = 3 Graph the polar equation. 44) r = 6 sin θ r Decompose v into two vectors v1 and v, where v1 is parallel to w and v is orthogonal to w. 4) v = i - 4j, w = -3i + j Test the equation for smmetr with respect to the given ais, line, or pole. 46) r = 4 sin θ; the pole Decompose v into two vectors v1 and v, where v1 is parallel to w and v is orthogonal to w. 47) v = i + 7j, w = i + j Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. 48) a = 8, b =, c = 4 Dr. James Math 41 PreCalculus 10/4/011 Page

6 Graph the parabola. 49) = ) The arch beneath a bridge is semi-elliptical, a one-wa roadwa passes under the arch. The width of the roadwa is 36 feet and the height of the arch over the center of the roadwa is 9 feet. Two trucks plan to use this road. The are both 8 feet wide. Truck 1 has an overall height of 9 feet and Truck has an overall height of 8 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the bridge. Find the focus and directri of the parabola with the given equation. 1) = 4 Identif the equation without appling a rotation of aes. ) = 0 Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described b the parametric equations corresponding to the given value of t. 3) = t3 + 1, = 6 - t4; t = Graph the ellipse and locate the foci. 4) 9 = Dr. James Math 41 PreCalculus 10/4/011 Page 6

7 Graph the parabola. ) = - - Find the foci of the ellipse whose equation is given. 6) 16( - ) + 36( + ) = 76 7) ( - 3) 9 + ( + 1) 36 = 1 Find the standard form of the equation of the hperbola satisfing the given conditions. 8) Endpoints of transverse ais: (-6, 0), (6, 0); foci: (-10, 0), (-10, 0) 9) (cot + 1)(cot + 1) - csc cot =? 60) tan (π - θ) =? Use the given information to find the eact value of the epression. 61) Find cos (α + β). sin α = 4, α lies in quadrant I, and cos β = 1, β lies in quadrant I. 13 Rewrite the epression in terms of the given function or functions. 6) csc + tan csc ; cos and sin 63) tan - (1 + tan ) =? 64) sin (α + β) + sin (α - β) =? 6) The distance from home plate to dead center field in Sun Devil Stadium is 400 feet. A baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it from first base to dead center field? Dr. James Math 41 PreCalculus 10/4/011 Page 7

8 Find the unit vector having the same direction as v. 66) v = -3i - 4j Use a polar coordinate sstem to plot the point with the given polar coordinates. 67) (-, 3π 4 ) Find another representation, (r, θ), for the point under the given conditions. 68) 1, π 4, r < 0 and 0 < θ < π Graph the polar equation. 69) r = 6 cos θ r Find the angle between the given vectors. 70) u = 4j, v = 9i - j Dr. James Math 41 PreCalculus 10/4/011 Page 8

9 Find the standard form of the equation of the hperbola. 71) Rewrite the epression as an equivalent epression that does not contain powers of trigonometric functions greater than 1. 7) cos3 73) sin + sin cot =? Use the given information given to find the eact value of the trigonometric function. 74) sin θ = 1 4, tan θ > 0 Find cos θ. Write the comple number in polar form. Epress the argument in degrees. 7) -4 Write the comple number in rectangular form. 76) 4(cos 11 + i sin 11 ) Epress answers relative to an ''-sstem in which the given equation has no ''-term. 77) = 0; Find the coordinates of the vertices on the minor ais. Find the eact value b using a sum or difference identit. 78) cos ( ) Polar coordinates of a point are given. Find the rectangular coordinates of the point. 79).0, π 9 Dr. James Math 41 PreCalculus 10/4/011 Page 9