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NAME DATE PERIOD7-1EnrichmentThe Physics of SoccerRecall from Lesson 7-1 that the formula for the maximum height hv02 sin2 of a projectile is h , where is the measure of the angle of2gelevation in degrees, v0 is the initial velocity in feet per second, and gis the acceleration due to gravity in feet per second squared.Solve. Give answers to the nearest tenth.1. A soccer player kicks a ball at an initial velocity of 60 ft/s and anangle of elevation of 40 . The acceleration due to gravity is 32ft/s2. Find the maximum height reached by the ball.2. With what initial velocity must you kick a ball at an angle of 35 in order for it to reach a maximum height of 20 ft?The distance d that a projected object travels is given by theformula d 2v02 sin cos g .3. Find the distance traveled by the ball described in Exercise 1.In order to kick a ball the greatest possible distance at a given2v2sin cos 0initial velocity, a soccer player must maximize d g .Since 2, v0, and g are constants, this means the player mustmaximize sin cos .sin 0 cos 0 sin 90 cos 90 0sin 10 cos 10 sin 80 cos 80 0.1710sin 20 cos 20 sin 70 cos 70 0.32144. Use the patterns in the table to hypothesize a value of for whichsin cos will be maximal. Use a calculator to check yourhypothesis. At what angle should the player kick the ball toachieve the greatest distance? Glencoe/McGraw-Hill277Advanced Mathematical Concepts

NAME DATE PERIOD7-2Study GuideVerifying Trigonometric IdentitiesWhen verifying trigonometric identities, you cannot add orsubtract quantities from each side of the identity. Anunverified identity is not an equation, so the properties ofequality do not apply.Example 1sec2 x 1 sin2 x is an identity.Verify that 2sec xSince the left side is more complicated,transform it into the expression on the right.sec2 x 1 sec2 x(tan2 x 1) 1 sec2 x2 xtan sec2 x2 xsin cos2 x 1 2cos x2 xsin cos2 xsin2 xsin2 xsec2 x tan2 x 1sin2 xSimplify.sin2 xx , sec2 x 1 sin tan2 x cos2 xcos2 x2 cos2 x sin2 xsin2 x sin2 xMultiply.The techniques that you use to verify trigonometric identitiescan also be used to simplify trigonometric equations.Example 2Find a numerical value of one trigonometricfunction of x if cos x csc x 3.You can simplify the trigonometric epression on the leftside by writing it in terms of sine and cosine.cos x csc x 31 3cos x sin xcos x 3 sin x1 csc x sin xcot x 3cos xcot x sin xMultiply.Therefore, if cos x csc x 3, then cot x 3. Glencoe/McGraw-Hill278Advanced Mathematical Concepts

NAME DATE PERIOD7-2PracticeVerifying Trigonometric IdentitiesVerify that each equation is an identity.csc x1. cos xcot x tan x1 1 2 sec2 y2. sin y 1sin y 13. sin3 x cos3 x (1 sin x cos x)(sin x cos x)cos sec 4. tan 1 sin Find a numerical value of one trigonometric function of x.5. sin x cot x 16. sin x 3 cos x7. cos x cot x8. Physics The work done in moving an object is given by the formulaW Fd cos , where d is the displacement, F is the force exerted, and is the angle between the displacement and the force. Verify thatcot is an equivalent formula.W Fd csc Glencoe/McGraw-Hill279Advanced Mathematical Concepts

NAME DATE PERIOD7-2EnrichmentBuilding from 1 1By starting with the most fundamental identity of all, 1 1, you cancreate new identities as complex as you would like them to be.First, think of ways to write 1 using trigonometric identities. Someexamples are the following.1 cos A sec A1 csc2 A cot2 A1 cos (A 360 ) cos (360 A)Choose two such expressions and write a new identity.cos A sec A csc 2 A cot 2 ANow multiply the terms of the identity by the terms of another identityof your choosing, preferably one that will allow some simplificationupon multiplication.cos A sec A csc 2A cot 2Asin A cos A tan A sin A sec A tan A csc 2 A cot ABeginning with 1 1, create two trigonometric identities.1.2.Verify that each of the identities you created is an identity.3.4. Glencoe/McGraw-Hill280Advanced Mathematical Concepts

NAME DATE PERIOD7-3Study GuideSum and Difference IdentitiesYou can use the sum and difference identities and thevalues of the trigonometric functions of common angles tofind the values of trigonometric functions of other angles.Notice how the addition and subtraction symbols are relatedin the sum and difference identities.Sum and Difference IdentitiesCosine functioncos ( ) cos cos sin sin Sine functionsin ( ) sin cos cos sin tan tan tan ( ) 1 tan tan Tangent functionExample 1Use the sum or difference identity for cosine tofind the exact value of cos 375 .375 360 15 cos 375 cos 15 Symmetry identity, Case 1cos 15 cos (60 45 ) 60 and 45 are two commonangles that differ by 15 .cos 15 cos 60 cos 45 sin 60 sin 45 Difference identity for cosine 2 3 or 2 2 6 cos 15 12 2224Example 2 , 0 y ,Find the value of sin (x y) if 0 x 22312.sin x 5 , and sin y 3 7In order to use the sum identity for sine, youneed to know cos x and cos y. Use a Pythagoreanidentity to determine the necessary values.sin2 cos2 1 cos2 1 sin2 .Pythagorean identitySince it is given that the angles are in Quadrant I,the values of sine and cosine are positive. Therefore, ins 2 .cos 1cos x 1 35 cos y 126 5 or 45 2 1 132 7 2 112 32 6 5 9 or 335 7Now substitute these values into the sumidentity for sine.sin (x y) sin x cos y cos x sin y15 3 35 3357 45 1327 or 185 Glencoe/McGraw-Hill281Advanced Mathematical Concepts

NAME DATE PERIOD7-3PracticeSum and Difference IdentitiesUse sum or difference identities to find the exact value of each trigonometricfunction.5 1. cos 122. sin ( 165 )3. tan 345 4. csc 915 7 5. tan 12 6. sec 12 and 0 y .Find each exact value if 0 x 227. cos (x y) if sin x 15 and sin y 45 38. sin (x y) if cos x 18 and cos y 35 73 and cot y 4 9. tan (x y) if csc x 15 3Verify that each equation is an identity.10. cos (180 ) cos 11. sin (360 ) sin 12. Physics Sound waves can be modeled by equations of theform y 20 sin (3t ). Determine what type of interferenceresults when sound waves modeled by the equationsy 20 sin (3t 90 ) and y 20 sin (3t 270 ) are combined.(Hint: Refer to the application in Lesson 7-3.) Glencoe/McGraw-Hill282Advanced Mathematical Concepts

NAME DATE PERIOD7-3EnrichmentIdentities for the Products of Sines and CosinesBy adding the identities for the sines of the sum and difference of themeasures of two angles, a new identity is obtained.sin ( ) sin cos cos sin sin ( ) sin cos cos sin (i) sin ( ) sin ( ) 2 sin cos This new identity is useful for expressing certain products as sums.ExampleWrite sin 3 cos as a sum.In the right side of identity (i) let 3 and sothat 2 sin 3 cos sin (3 ) sin (3 ).11Thus, sin 3 cos sin 4 sin 2 .22By subtracting the identities for sin ( ) andsin ( ), you obtain a similar identity for expressing aproduct as a difference.(ii) sin ( ) sin ( ) 2 cos sin (sin 3x sin x)cos 2x sin x .Verify the identity sin 2x cos xsin2 3x sin2 x2ExampleIn the right sides of identities (i) and (ii) let 2x and x. Then write the following quotient.2 cos 2x sin xsin (2x x) sin (2x x) 2 sin 2x cos xsin (2x x) sin (2x x)By simplifying and multiplying by the conjugate, theidentity is verified.cos 2x sin xsin 3x sin x sin 3x sin x . sin 3x sin xsin 2x cos x sin 3x sin x(sin 3x sin x)2 sin2 3x sin2 xComplete.1. Use the identities for cos ( ) and cos ( ) to find identities for expressing theproducts 2 cos cos and 2 sin sin as a sum or difference.2. Find the value of sin 105 cos 75 by using the identity above. Glencoe/McGraw-Hill283Advanced Mathematical Concepts

NAME DATE PERIOD7-4Study GuideDouble-Angle and Half-Angle IdentitiesExample 1If sin 41 and has its terminal side in the firstquadrant, find the exact value of sin 2 .To use the double-angle identity for sin 2 , we mustfirst find cos .sin2 cos2 12 41 cos2 1sin 41 15 cos2 16 1 5 cos 4Now find sin 2 .sin 2 2 sin cos 1 5 2 41 4Double-angle identity for sine 1 5 sin 14 , cos 4 1 5 8Example 2Use a half-angle identity to find the exact value.of sin 1 2sin 1 2 sin 6 21 cos 6 2 c 1 os aUse sin 2 . Since 1 2 is in2Quadrant I, choose the positive sine value. 31 2 2 3 2 4 2 3 2 Glencoe/McGraw-Hill284Advanced Mathematical Concepts

NAME DATE PERIOD7-4PracticeDouble-Angle and Half-Angle IdentitiesUse a half-angle identity to find the exact value of each function. 2. tan 81. sin 105 5 3. cos 8Use the given information to find sin 2 , cos 2 , and tan 2 .4. sin 112 , 0 3 6. sec 52 , 2 90 5. tan 12 , 7. sin 35 , 0 3 2 2Verify that each equation is an identity.8. 1 sin 2x (sin x cos x)2sin 2x9. cos x sin x 210. Baseball A batter hits a ball with an initial velocity v0 of100 feet per second at an angle to the horizontal. An outfieldercatches the ball 200 feet from home plate. Find if the rangev 2 sin 2 .of a projectile is given by the formula R 31 2 0 Glencoe/McGraw-Hill285Advanced Mathematical Concepts

NAME DATE PERIOD7-4EnrichmentReading Mathematics: Using ExamplesMost mathematics books, including this one, use examples to illustratethe material of each lesson. Examples are chosen by the authors toshow how to apply the methods of the lesson and to point out placeswhere possible errors can arise.1. Explain the purpose of Example 1c in Lesson 7-4.2. Explain the purpose of Example 3 in Lesson 7-4.3. Explain the purpose of Example 4 in Lesson 7-4.To make the best use of the examples in a lesson, try following thisprocedure:a. When you come to an example, stop. Think about what you havejust read. If you don’t understand it, reread the previous section.b. Read the example problem. Then instead of reading the solution,try solving the problem yourself.c. After you have solved the problem or gone as far as you can go,study the solution given in the text. Compare your method andsolution with those of the authors. If necessary, find out whereyou went wrong. If you don’t understand the solution, reread thetext or ask your teacher for help.4. Explain the advantage of working an example yourself oversimply reading the solution given in the text. Glencoe/McGraw-Hill286Advanced Mathematical Concepts

NAME DATE PERIOD7-5Study GuideSolving Trigonometric EquationsWhen you solve trigonometric equations for principal values ofx, x is in the interval 90 x 90 for sin x and tan x. Forcos x, x is in the interval 0 x 180 . If an equation cannotbe solved easily by factoring, try writing the expressions interms of only one trigonometric function.Example 1Solve tan x cos x cos x 0 for principal valuesof x. Express solutions in degrees.tan x cos x cos x 0cos x (tan x 1) 0Factor.cos x 0 or tan x 1 0Set each factor equal to 0.x 90 tan x 1x 45 When x 90 , tan x is undefined, so the only principalvalue is 45 .Example 2Solve 2 tan2 x sec2 x 3 1 2 tan x for 0 x 2 .This equation can be written in terms of tan x only.2 tan2 x sec2 x 3 1 2 tan x2 tan2 x (tan2 x 1) 3 1 2 tan x sec2 x tan2 x 1tan2 x 2 1 2 tan x Simplify.tan2 x 2 tan x 1 0Factor.(tan x 1)2 0tan x 1 0Take the square root of each side.tan x 13 7 or x x 44When you solve for all values of x, the solution should berepresented as x 360 k or x 2 k for sin x and cos x andx 180 k or x k for tan x, where k is any integer.Example 3 sin x for all real values of x.Solve sin x 3sin x 3 sin x 02 sin x 3 2 sin x 3 3 sin x 2 4 5 2 k or x 2 k, where k is any integerx 334 5 2 k and 2 k.The solutions are 33 Glencoe/McGraw-Hill287Advanced Mathematical Concepts

NAME DATE PERIOD7-5PracticeSolving Trigonometric EquationsSolve each equation for principal values of x. Expresssolutions in degrees.1. cos x 3 cos x 22. 2 sin2 x 1 0Solve each equation for 0 x 360 .4. cos 2x 3 cos x 1 03. sec2 x tan x 1 0Solve each equation for 0 x 2 .6. cos 2x sin x 15. 4 sin2 x 4 sin x 1 0Solve each equation for all real values of x.7. 3 cos 2x 5 cos x 18. 2 sin2 x 5 sin x 2 09. 3 sec2 x 4 010. tan x (tan x 1) 011. Aviation An airplane takes off from the ground and reachesa height of 500 feet after flying 2 miles. Given the formulaH d tan , where H is the height of the plane and d is thedistance (along the ground) the plane has flown, find the angleof ascent at which the plane took off. Glencoe/McGraw-Hill288Advanced Mathematical Concepts

NAME DATE PERIOD7-5EnrichmentThe SpectrumIn some ways, light behaves as though it were composed of waves. Thewavelength of visible light ranges from about 4 10 5 cm for violetlight to about 7 10 5 cm for red light.As light passes through a medium, its velocity depends upon thewavelength of the light. The greater the wavelength, the greater thevelocity. Since white light, including sunlight, is composed of light ofvarying wavelengths, waves will pass through the medium at aninfinite number of different speeds. The index of refraction n ofcthe medium is defined by n v , where c is the velocity of light in avacuum (3 1010 cm/s), and v is the velocity of light in the medium.As you can see, the index of refraction of a medium is not a constant. Itdepends on the wavelength and the velocity of light passing through it.(The index of refraction of diamond given in the lesson is an average.)1. For all media, n 1. Is the speed of light in a medium greater thanor less than c? Explain.2. A beam of violet light travels through water at a speed of2.234 1010 cm/s. Find the index of refraction of water for violetlight.The diagram shows why a prism splits white lightinto a spectrum. Because they travel at differentvelocities in the prism, waves of light of differentcolors are refracted different amounts.3. Beams of red and violet light strike crown glass at an angle of20 . Use Snell’s Law to find the difference between the angles ofrefraction of the two beams.violet light: n 1.531red light: n 1.513 Glencoe/McGraw-Hill289Advanced Mathematical Concepts

NAME DATE PERIOD7-6Study GuideNormal Form of a Linear EquationNormal FormThe normal form of a linear equation is x cos y sin p 0,where p is the length of the normal from the line to the origin and is the positive angle formed by the positive x-axis and the normal.You can write the standard form of a linear equation if youare given the values of and p.Example 1Write the standard form of the equation of a linefor which the length of the normal segment tothe origin is 5 and the normal makes an angle of135 with the positive x-axis.x cos y sin p 0 Normal formx cos 135 y sin 135 5 0 135 and p 5 x 2 y 5 02 22 2 x 2 y 10 0 Multiply each side by 2.The standard form of the equation is 2 x 2 y 10 0.The standard form of a linear equation, Ax By C 0, can bechanged to the normal form by dividing each term of the equationby A 2 B2 . The sign is chosen opposite the sign of C. You canthen find the length of the normal, p units, and the angle .Example 2Write 3x 4y 10 0 in normal form. Then findthe length of the normal and the angle it makeswith the positive x-axis. 2 B2 to determineSince C is negative, use Athe normal form. 2 B2 3 2 42 or 5 AThe normal form is 35 x 45 y 150 0 or 35 x 45 y 2 0.Therefore, cos 35 , sin 45 , and p 2.Since cos and sin are both positive, mustlie in Quadrant I.tan 45 35 or 43 sin tan cos 53 The normal segment has length 2 units andmakes an angle of 53 with the positive x-axis. Glencoe/McGraw-Hill290Advanced Mathematical Concepts

NAME DATE PERIOD7-6PracticeNormal Form of a Linear EquationWrite the standard form of the equation of each line, givenp, the length of the normal segment, and , the angle thenormal segment makes with the positive x-axis.1. p 4, 30 , 4 2. p 2 23. p 3, 60 5 4. p 8, 67 5. p 2 3 , 46. p 15, 225 Write each equation in normal form. Then find the length ofthe normal and the angle it makes with the positive x-axis.7. 3x 2y 1 08. 5x y 12 09. 4x 3y 4 010. y x 511. 2x y 1 012. x y 5 0 Glencoe/McGraw-Hill291Advanced Mathematical Concepts

NAME DATE PERIOD7-6EnrichmentSlopes of Perpendicular LinesThe derivation of the normal fo

Feb 02, 2015 · Page A1 is an answer sheet for the SAT and ACT Practice questions that appear in the Student Edition on page 483. Page A2 is an answer sheet for the SAT and ACT Practice master. These improve students’familiarity with the answer formats they may encounter in test taking. The ans