Math 2412 Activity 4(Due with Final Exam)

Transcription

1 Math Activity (Due with Final Exam) Use properties of similar triangles to find the values of x and y x y 7 7 x 5 x y 7 For the angle in standard position with the point 5, on its terminal side, find the values of the six trigonometric functions: 5, sin cos tan csc sec cot

2 3 Find one solution of the equation sin 0 cos3 0 {Hint: cos x sin90 x } Find all the trigonometric function values of, if csc and is in Quadrant III sin cos tan csc sec cot 5 Find the exact value of each labeled part: a q m n q 5 n 60 m a 7 6 Find all the exact trigonometric function values of 590 sin590 cos590 tan590 csc590

3 sec590 cot590 7 Solve the right triangle to the nearest tenth of a degree and tenth of a foot: A b 895 ft C a 79 B m A b a 8 Solve the right triangle to the nearest degree and the nearest foot: A 37 ft c C 56 ft B

4 9 Find h to the nearest tenth h 35 {Hint: cot 35 x x h and cot 35 x35 x 35 } h h h 0 Find h to the nearest tenth x h {Hint: cot 35 x and cot 35 x h x h } Find the area of the indicated sector: 5 8

5 Find the measure of the central angle,, in radians The rotation of the larger wheel causes the smaller wheel to rotate Find the radius of the larger wheel if the smaller wheel rotates 90 when the larger wheel rotates 60 ft r Graph the function y cos x on the interval 0,

6 5 Graph the function y 3 sin x on the interval 3 0, Graph the function y3sin x 3 on the interval 0, 3 5

7 7 Graph the function y3cosx on the interval, 8 8

8 sin Graph the function y x on the interval,

9 7 9 Graph the function y3secx on the interval, Graph the function y csc x on the interval 3, 3

10 sec Graph the function y x on the interval, x Graph the function y tan on the interval 5, 3 5 3

11 3 Graph the function y 3cot x on the interval, Graph the function y tan x on the interval 3 3,6

12 5 Determine the range of the following functions: a) y 3sin x 7 b) y x sec 8 6 Verify the identity 7 Verify the identity cos x sin x cos x tan x sec x sec x cos sin sec cos x x x x 8 Show that the equation cosx cos x sin x is not an identity by demonstrating that for a specific value of x it is false 9 Show that the equation sinx sin x cos x is not an identity by demonstrating that for a specific value of x it is false 30 Find the exact value of cos65 3 Find an exact value of that makes cot 0 tan 0 3 Verify the identity cos 90 sin sin cos true x x x x 33 Find the exact value of cos cos9 sin sin9 3 Find the exact value of tan tan tan tan Find the exact value of sin65 36 Verify the identity tan x y tan y x tan x tan y tan xtan y 37 Verify the identity sin x y cot x cot y cos x y cot xcot y 38 Find the exact value of cos sin 39 Find the exact value of sin5 cos5 0 Verify the identity cos x cot x sin x

13 tan Verify the identity tan x x cosx Find the exact value of cot, if x 3 Verify the identity tan csc x cot x 5 tan and Verify the identity tan cos x tan x x 5 Find the exact value of 6 Find the exact value of 7 Find the exact value of 8 Find the exact value of sin sec 3 tan cos cos sin 3 5 cos Find the exact value of 50 Solve the equation 5 Solve the equation sin sin 3 cos cos 0 on the interval sin 0 on the interval 0, 0, 5 Solve the equation 53 Solve the equation sec tan on the interval cos sin 0, on the interval 0, 5 Solve the equation cos x 0 on the interval 0, 55 Solve the equation cos x sin x 0 on the interval 0, x 56 Solve the equation tan sin x on the interval 0,

14 57 Solve the equation sin cos x x on the interval 0, Sketch the solutions of the following polar coordinate equations 58 r sin 59 r cos 60 r cos 6 r cos 6 r cos Find the points of intersection of the solution curves of the following pairs of polar coordinate equations 63 r cos, r cos 6 rcos3, r

15 Find the points of intersection of the curves defined by the following parametric equations 65 xt yt ; 3 t and xs ; 3 s ys 66 x cost ;0 t y 3sint x sec s and ; 3 s 3 y tan s

16 67 x cost y sin t ;0 t and x cos s ;0 s y sin s 68 Find the exact value of each part labeled with a variable 8 y w x z 69 The tires of a bicycle have a radius of 5 ft, and are turning at the rate of 5 revolutions per second How fast is the bicycle traveling in feet per second?

17 70 If tan x 75 and cos 8 7 Find the exact value of cos x, then find the value of tan x cos x {Hint: 3 and cos A B cos Acos B cos Acos B } 7 Find the exact value of 5 tan {Hint: 5 6 and tan A B tan A tan B } tan Atan B 73 Find the exact value of cos A cos A {Hint: cos and 6 } Find the exact value of the following: 7 sinsin 75 sin sin 3 76 cossin 3 77 sin tan 78 tancos For each of the following, find sin x y, cos x y, tan x y, and the quadrant of x y sin x, cos y, x in quadrant I, y in quadrant IV 0 5 sin y, cos x, x in quadrant II, y in quadrant III 3 5

18 Find the sine and cosine of the following 8 B, given cosb, B in quadrant IV 8 y, given 8 5 sec y, sin y 0 3 Find the following: A 83 sin, given 3 cos A, with 90 A 80 b) sin x, given sin x 6, with x 8 sin y, given cosy, with y 3 Exactly solve the following trigonometric equations on the interval 0, 85 sin x 86 3cos x cos x 0 87 x x sec x 88 csc sin sin x sinx 90 cosxcos x 0 9 sin x cos x 9 sin3x 0 93 cos x 9 6 sin x cos x 95 6sin x7sin x 0 96 Sketch the graph of the solution to the polar coordinate equation r sin r

19 97 Sketch the graph of the solution to the polar coordinate equation r cos r 3 98 Find the points of intersection of the solution curves of the polar coordinate equations r cos and r sin 99 Find the points of intersection of the solution curves of the polar coordinate equations r sin and r sin cos

20 00 Graph the function y tan x on the interval, 0 Graph the function y sinx on the interval 0, 0 Determine the range of the function y x 03 If 3 8sin 5 7 cos x, then find the exact value of sin xtan x sin xcot x Find the exact value of the following 0 sincos sin sin tan sin 3 sin 3 {Hint: {Hint: sin A sin Acos A} sin A B sin Acos B cos Asin B } {Hint: tan A sin A cos A } cos A sin A 07 cos sin 08 Sketch the graph of the solution to the polar coordinate equation r cos r

21 09 Sketch the graph of the solution to the polar coordinate equation r sin r Find the points of intersection of the solution curves of the polar coordinate equations r sin and r 3sin

22 Find the points of intersection of the solution curves of the polar coordinate equations r sin and r Find the area of the region that is inside the solution curve of r sin but outside the solution curve of r sin 3 Given that a i 3j and b i j and another vector r 6i 7j, find numbers k and m so that r ka mb Express c in terms of a and b, given that the tip of c bisects the line segment b a c 5 For what values of x are xi j and xi xj orthogonal?

23 6 Given that a i xj and b i yj, find all values of x and y so that a b and a b 7 Use the dot-product to show that an angle inscribed in a semi-circle is a right angle (Look at a b a b ) a b a a b b b 8 Show that the sum of the squares of the lengths of the diagonals of a parallelogram equals the sum of the squares of the lengths of the four sides a Expand a b a b by using the dot-product b a b b a b a 9 It looks as if a b and a b are orthogonal Is this mere coincidence, or are there circumstances where we would expect the sum and difference of two vectors to be orthogonal? Find out by expanding a b a b 0 a b b a b a b

24 0 Given vectors a and b, let m a and n b, show that a) na mb and na mb are orthogonal b) c na mb bisects the angle between a and b Find all vectors v in the plane so that v and vi Graph each parabola x y 3 y x x 8x 8y Graph each ellipse 5 x y x 6y x 5y x 8x y 8y 3 0 Graph each hyperbola 9 x y y x 3 x 5y x 8x y 8y Find an equation for the parabola with focus of, and directrix of y

25 3 Find an equation of the hyperbola satisfying the given conditions: Endpoints of transverse axis:,0,,0 ; asymptote y x 35 Solve the system x x y 9 y 9 Solve the following systems of equations Check to see if your answer agrees with the graph x y (line) 36 y x (parabola) 37 x y 5 (circle) 3x y 5 (line)

26 x y (hyperbola) 38 x y (ellipse) x y (ellipse) 3 5 x y (hyperbola) 0 y x x (parabola) y x (parabola) y x (parabola) x y 6 (ellipse)

27 Find the values of x and y in the figure x 0 7 y 9 3 Express the product of the following complex numbers in standard form a) z 3cos00 i sin00, w cos60 i sin 60 b) z cos0 isin0, w 6cos5 isin5 Express the following in standard form a) 3 3 cos80 isin80 b) 5 cos isin On a recent episode of Who Wants to Be a Millionaire with Cedric the Entertainer, the following question appeared For which of the following times will the minute and hour hands of a clock form a right angle? a) :05 b) 5:0 c) 3:35 d) :50 The contestant chose answer a) and he was told that he was correct He wasn t correct, in fact, none of the options are correct Let s use basic trigonometry to find all the times for which the minute and hour hands form a right angle For t measured in minutes after M t t H t t midnight, 30 represents the cumulative angle of the minute hand, and 360 represents the cumulative angle of the hour hand In order for the two hands to form a right angle, the difference between the cumulative angle of the minute hand and the cumulative angle of the hour hand must be an odd multiple of So we get that

28 t n n n M t H t n ; n,, t t n ; n,, ;,, 80 t ; n,, a) Use the previous formula to find the number of times from one midnight to the next that the minute and hour hands form a right angle {Hint: 80 n # of minutes in a hour period } b) Use the same reasoning to find a formula for the times(in minutes after midnight) from one midnight to the next(inclusive) that the minute and hour hands point in exactly the same direction, and the number of times that it occurs c) Use the same reasoning to find a formula for the times(in minutes after midnight) from one midnight to the next that the minute and hour hands point in exactly opposite directions, and the number of times that it occurs

29 6 a) Use geometry to fill in all the missing angles and sides in the following diagram of right triangle ABC inscribed inside rectangle ADEF A 5 30 F C 90 D 60 B 90 E b) Use the diagram to find the exact values of the sine, cosine, tangent, cotangent, secant, and cosecant of the angles 5 and 75