MATH 9 TRIGONOMETRY COURSE PACK (Fall 08) Mark Turner Mathematics Division Cuesta College
Angles and Triangles. Find the complement and supplement of 60. Complement = Supplement =. Use the Pythagorean Theorem to solve the right triangle for x. x 5 x 3. The longest side in a 30-60 -90 triangle is 8. Find the lengths of the other two sides. 4. Given θ = 40. (a) Draw θ in standard position. (b) Find a positive and negative angle coterminal with θ. (c) Find all angles coterminal with θ. (d) Find and label a point on the terminal side of θ, and then find the distance of that point from the origin.
Trig Ratios and Basic Identities. Draw 90 in standard position, find a point on the terminal side, and state the sine, cosine, and tangent. sin 90 cos90 tan 90. As increases from 0 to 90, the value of cos tends toward what number? 3. Complete the following table, giving the sign (+ or ) of each trig ratio in each quadrant. QI QII QIII QIV sine, cosecant cosine, secant tangent, cotangent 4. Indicate the two quadrants could terminate in if sin and tan have the same sign. 5. If sin /3 and cos 5 /3, find: cot csc
Trigonometric Identities. Simplify: sec tancsc. Multiply: (cos ) 3. Prove: sec tan csc 4. Prove: sin sec cos cos 5. Prove: (sin cos ) sin cos
Trigonometric Ratios Exact Values First, complete each figure by filling in the appropriate values. (, ) (, ) Now use the above figures to complete the table using exact values. You do not need to rationalize any denominators. sin cos tan 0 30 45 60 90 Use the values in the table above to find the exact values for the remaining trigonometric ratios. csc sec cot 0 30 45 60 90
DD, DMS, and Solving Right Triangles. Add: 63 38 and 4 5. Evaluate sin and csc for = 684' 3. Find acute angle such that cos = 0.9968. Round to the nearest tenth of a degree. 4. If A = 3, BDC = 48, AB = 7, and DB =, find x and y.
Math 9 Right Triangle Applications. An airplane takes off at an angle of 00 and flies at this angle until it reaches an altitude of 5,00 feet. How far, in miles, has the airplane traveled horizontally? (Assume the flight began at sea level.). Two forest fire lookouts are on a north-south shoreline, 7. miles apart. The bearing of a fire from lookout point A is S7.0W and from lookout point B is N63.0W. How far from lookout B is the fire? 3. A television tower stands on top of a building. From a point 75.3 feet from the base of the building, the angles of elevation to the top and base of the tower are 60. and 47.4, respectively. How tall is the tower? 4. Mark stands due south of the Eiffel Tower and notes that the angle of elevation to its top is 59.6. His friend John, standing 98 feet due east of Mark, notes an angle of elevation of 56.9. How tall is the tower? 5. Two Coast Guard lookouts, A and B, are on an east-west line.3 km apart. The bearing of a ship from lookout A is N3.0W and the bearing from lookout B is N65.0E. How far is the ship from lookout B?
Math 9 Vector Applications. A girl pulls a sled through the snow by exerting a force of 0 lb at an angle of 40 with the horizontal. Find the magnitude of the horizontal and vertical vector components of the force.. A sailboat travels at a constant speed. In two hours, the sailboat has traveled 0 miles north and 4 miles east. What is its velocity? 3. A weight of 000 lb is supported by two metal braces as shown below. Find the magnitudes of the tension forces in the direction of each of the braces. B A 35Þ 35 C 000 lb 4. Luke and Logan are pulled on a sled 60 feet through the snow by their dad, who pulls with a force of 36 pounds at an angle of 40 with the horizontal (above right). How much work is done?
REFERENCE ANGLE AND TRIANGLE Assume P is a point on the terminal side of and that 0 < 360 ˆ ˆ QII: ˆ 80 QI: ˆ ˆ ˆ QIII: ˆ 80 QIV: ˆ 360
Reference Angles. Use a reference angle to find sin 5.. Use a reference angle to find sec330. 3. Use a reference angle to find cos( 35 ). 4. Find θ, to the nearest tenth of a degree, if QII and tan 0.084.
Radian Measure. Convert 4 radians to degrees. Special Angles in Radians. Convert each angle to radians (exact values). Degrees 0 30 45 60 90 80 70 360 Radians 3. Use a reference angle to find 7 cot. 4 4. Evaluate 4 sin(3 x ) when 3 x. 6
Functions. Find six ordered pairs that satisfy the relation x y domain and range for this relation., then use them to sketch the graph. State the x y Domain = Range =. Determine if the equation defines y as a function of x. Explain why or why not. 3 (a) x y (b) y x 3. Given f ( x) x 3x, find: f(0) = f(4) = f() = 4. Graph the function x f ( x) x x by finding six ordered pairs and plotting points. f(x)
5. Use the graph of y f( x) shown to find or approximate the indicated values. f(0) = f(3) f() =
Circular Functions. Use the unit circle shown to estimate θ ( 0 ), such that sin 0.4.. Consider cos(5.5) : (a) What is the function? (b) What is the argument of the function? (c) What is the value of the function? 3. Is the statement csc(0) z possible for some number z? Explain why or why not. 4. Is the statement csc( z) 0 possible for some number z? Explain why or why not. 5. Explain how the value of each trigonometric function varies as θ increases from 0 to /. (a) sin (b) cos (c) tan
Arc Length and Sector Area. Find the arc length if 60 and r = 4 mm.. From the earth, the sun subtends an angle of approximately 0.5. If the distance to the sun is about 93 million miles, estimate the diameter of the sun accurate to the nearest ten-thousand miles. 93,000,000 mi 3. An automobile windshield wiper 0 inches long rotates through an angle of 60. If the rubber part of the blade covers only the last 9 inches of the wiper, find the area of the windshield cleaned by the windshield wiper.
Circular Functions First, complete the table by copying down the coordinates of point P on the unit circle. t x y t x y 0 4.5 0.5 5.0.0 5.5.5 6.0.0.5 / 3.0 3.5 3 / 4.0. Plot y against t to draw the graph of the sine function. y 3 6 t. Plot /y against t to draw the graph of the cosecant 3. Plot y/x against t to draw the graph of function. the tangent function. /y y/x 3 6 t 3 t
Math 9 THE SIX BASIC CYCLES y = sin x y = cos x 3 3 - - 4 3 y = csc x y = sec x 3 4 3 - - -3-4 - - -3-4 3 y = tan x y = cot x 4 3 4 3 3 4 3 4 - - 4 - - 4-3 -4-3 -4
Graphs of Sine and Cosine. Graph y cos x and y 3cosx. 3 4. Graph y cos x and y cos( x). 3 4 3. Graph y 3cos(4 x ).
More Graphs and Modeling x. Graph y 3cot 8.. Find an equation to model the graph using a sine function and using a cosine function.
MODELING ANNUAL TORNADO DATA TOTAL ANNUAL YEAR TORNADOES (U.S.) 986 765 987 656 988 70 989 856 990 33 99 3 99 97 993 73 994 08 *Source: Storm Prediction Center (http://www.spc.noaa.gov/archive/tornadoes/ustdbmy.html)
Inverse Functions. If R = {(, ), (0, ), (, 3), (, )}, find: (a) Domain of R: (b) Range of R: (c) Is R a function? Why or why not? (d) Inverse of R = (e) Domain of inverse: (f) Range of inverse:. Sketch the graph of the inverse of y = f(x). y = f(x) (0.5, ) (, 0.5) Every relation, and therefore every function, has an inverse relation. Under what conditions will the inverse of a function be a function as well? 3. Both f and g shown below are functions. Determine if the inverse of each function is also a function. Explain why or why not. x f(x) 3 0 3 3 4 4 0 x g(x) 0 3 3 0 4
4. Determine if the function is one-to-one. Explain why or why not. 3 (a) f ( x) x (b) gx ( ) x
Inverse Trigonometric Functions. Graph the inverse cosine function y cos x. y cos x x 0 4 3 4 y y cos x x y. Evaluate tan 3 in both radians and degrees. 3. Use your calculator to approximate each of the following (in degrees). (a) Arctan(9.304) (b) sin.304
INVERSE TRIGONOMETRIC FUNCTIONS Definitions and Graphs DEFINITIONS y = sin x y = arcsin x are equivalent to x= sin y, where x and y y = cos x y = arccos x are equivalent to x= cos y, where x and 0 y y = tan x y = arctan x are equivalent to x= tan y, where < x < and < y < GRAPHS y = sin x y = cos x y = tan x - - - Domain: [, ] Domain: [, ] Domain: All real numbers Range:, Range: [ 0, ] Range:,
Proving Identities Prove each identity using column format and proper notation by transforming one side into the other side.. Prove: sec tan sin csc. Prove: cscx sec x cot x cos x 3. Prove: tan sec sec
Derivations of Identities Given: cos( A B) cos Acos B sin Asin B. cos( AB) cos( A( B)) cos A sin A. sin( A B) cos ( AB) 3. sin( AB) sin( A( B)) 4. sin( A B) tan( AB) cos( A B) 5. tan( AB) tan( A( B)) 6. sin( x) sin( xx)
7. cos( x) cos( xx) But sin x cos x and cos x sin cos( x) cos( x) x Isolating cos x : Isolating sin x : 8. tan( x) tan( xx) 9. Since cosa sin A sin A and cosa cos A cos A x sin x cos 0. x sin( x/ ) tan cos( x / ) But cos( A ) sin( A) x tan
TRIGONOMETRIC IDENTITIES RECIPROCAL IDENTITIES sin x = cos x = tan x = csc x sec x cot x csc x = sec x = cot x = sin x cos x tan x PYTHAGOREAN IDENTITIES RATIO IDENTITIES sin x cos x tan x = cot x = cos x sin x NEGATIVE ANGLE IDENTITIES sin( x) = sin x cos( x) = cos x tan( x) = tanx sin x + cos x = sin x = cos x sin x =± cos x tan x + = sec cot x + = csc cos x = sin x x x cos x =± sin x sin tan sec COFUNCTION IDENTITIES ( x) = cos x cos ( x ) ( x) = cot x cot ( x ) ( x) = csc x csc ( x ) = sin x = tan x = sec x SUM/DIFFERENCE IDENTITIES sin( x + y) = sin xcos y+ cos xsin y cos( x + y) = cos xcos y sin xsin y sin( x y) = sin xcos y cos xsin y cos( x y) = cos xcos y+ sin xsin y tan x + tan y tan( x+ y) = tanx tan y tan x tan y tan( x y) = + tanx tan y DOUBLE-ANGLE IDENTITIES tanx sin x = sin xcos x tan x = tan x cos x sin x cos x= sin x sin x = ( cos x) cos x cos x = ( + cos x) PRODUCT-SUM IDENTITIES [ ] sin x cos y = sin( x+ y) + sin( x y) [ ] cos xsin y = sin( x+ y) sin( x y) [ ] sin xsin y = cos( x y) cos( x+ y) [ ] cos x cos y = cos( x+ y) + cos( x y) HALF-ANGLE IDENTITIES x cosx sin =± x + cosx cos =± x cos x cos x sin x tan =± = = + cos x sin x + cos x SUM-PRODUCT IDENTITIES x + y x y sin x+ sin y = sin cos x + y x y sin x sin y = cos sin x + y x y cos x+ cos y = cos cos x + y x y cos x cos y = sin sin
Using The Identities. Prove: sin( ) sin.. Graph y cos3xcos x sin 3xsin x for 0 x. 3. Graph at least one cycle for y cos x. 4. Use a half-angle identity to find the exact value of tan( / 8).
sin 5. Use half-angle identities to prove: sin. cos
Solving Trigonometric Equations. Solve cos xsin xcos x 0 for: (a) 0 x ; (b) all radian solutions.. Solve sin cos for 0 360. 3. Solve tan4x 3 for: (a) all radian solutions; (b) 0 x.
Math 9 Law of Sines/Cosines Applications. A woman entering an outside glass elevator on the ground floor of a hotel glances up to the top of the building across the street and notices that the angle of elevation is 58. She rides the elevator up four floors (40 feet) and finds that the angle of elevation to the top of the building across the street is. How tall is the building across the street?. A 00 lb weight hangs from two wires attached to the ceiling at angles of 50 and 3 (from the horizontal). Find the magnitude of the tension in each wire. 50 50Þ 3 3Þ 3. A cruise ship sets a course N47E from an island to a port on the mainland, which is 50 miles away. After moving through strong currents, the ship is off course at a position P that is N33E and 80 miles from the island. Approximately how far is the ship from the port? In what direction should the ship head to correct its course? N P 80 mi 33Þ 33 50 mi 4. A cm piston rod joins a piston to a 4. cm crankshaft. What is the shortest distance d of the base of the piston from the center of the crankshaft when the rod makes an angle of 5.0? 4. cm d cm 5.0 5Þ
Oblique Triangles. Two lookout posts, A and B, which are located.4 miles apart on a north-south coastline, spot an illegal foreign fishing boat within the three mile limit. If post A reports the ship at S 37.5 W, and post B reports the same ship at N 9.7 W, how far is the ship from post A?. A plane flies at a constant ground speed of 450 miles per hour due east, and encounters a 50 mileper-hour wind from the northwest. Find the airspeed and heading that will allow the plane to maintain its ground speed and eastward direction. 3. A plane is traveling 80 miles per hour with heading 00. The wind currents are moving with constant speed in the direction 40. If the ground speed of the plane is 65 miles per hour, what is its true course?
More Vectors and Dot Product. Draw the vector V that goes from the origin to the point (4, 6). Then write V in component form, and find V. Component form: V = V =. Write V from the above problem in vector component form. 3. If u = i 5j and v = 3i + j, find 3u + v. 4. If u = i 5j, find a vector w such that u and w are orthogonal. (Hint: let w = ai + bj.)
Complex Numbers and Trigonometric Form. Find 4i.. Express z 5 in trigonometric form. 3. Express 4 i in trigonometric form. Approximate θ to the nearest tenth of a degree. 4. Express 3 cis 33.8 in standard a + bi form.
Operations with Complex Numbers in Trig Form For the following problems, leave your answers in trigonometric form.. Find zz and z/ z if z 5 cis 04 and z cis 34.. Find 3 ( cis 34 ). 3. Find the two square roots of 9 cis 50 and graph them.
Polar Coordinates. Plot (, 45 ), then give two other ordered pairs that name the same point.. Convert to rectangular coordinates. (a) (3, 50 ) (b) (, 5 ) 3. Convert to polar coordinates with r 0 and 0 360. (a) (0, 5) (b) ( 3, ) 4. Convert to polar form. Then isolate r. (a) x y 9 0 (b) x y 4x 5. Convert to rectangular form. (a) r 4cos (b) r 4sin
Graphs in Polar Coordinates. Graph the equation r 6cos by making a table and plotting points. θ r. Graph the equation r 3 3sin by first graphing on a rectangular coordinate system, and then transferring this to a polar coordinate system. r 80 360