UNIT 10 Trigonometry UNIT OBJECTIVES 287

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1 UNIT 10 Trigonometry Literally translated, the word trigonometry means triangle measurement. Right triangle trigonometry is the study of the relationships etween the side lengths and angle measures of right triangles. Trigonometry is one of the most useful sujects in all of geometry ecause of its many real-world applications. stronomy, medical imaging, meteorology, cartography, and computer graphics are just a few of the fields in which it is used. Trigonometry is also very useful in finding indirect measurements. During The Great Trigonometric Survey of the 1800s, trigonometry was used to find land measurements necessary for mapmaking. Indirect measurement was used to find the heights of these Himalayan mountains: K2, Kanchenjunga, and Mount Everest. In this unit, you will study the three asic trigonometric ratios sine, cosine, and tangent and use them to solve prolems involving oth right triangles and all other triangles. UNIT OJETIVES Define the sine, cosine, and tangent ratios. Use the sine, cosine, and tangent ratios to find missing angle measures and missing side lengths in right triangles. Identify trigonometric identities. Use the relationships in and triangles to find trigonometric ratios and use those ratios to solve prolems. Use the Laws of Sines and osines to find missing angle or side measures in triangles. unit 10 TRIGONOMETRY 287

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3 Tangents Ojectives Define the tangent ratio and epress tangent ratios as fractions or decimals. Use a calculator to find the value of the tangent of an angle. Sometimes guideooks give the grade of a hiking trail. What does it mean when a hill has a grade of 15%? It means the slope is 15%, so it rises 15 feet over a horizontal distance of 100 feet. ut that can e hard to comprehend. Knowing the angle of elevation might give you a etter idea of how steep and difficult you can epect your clim to e. The first trigonometric ratio you will learn aout, tangent, will show you how you can find the angle of elevation of a hill with a given grade. Find the measure of an angle, given its tangent value. KEYWORDS adjacent side opposite side trigonometric ratio inverse tangent tangent Trigonometric Ratios When you find the ratio of the lengths of two sides of a right triangle, you are finding a trigonometric ratio. These ratios relate the sides of the triangle to either of the acute angles. Specific to one of the acute angles, the sides are referred to as adjacent, opposite, and hypotenuse. The hypotenuse is the side opposite the right angle. The opposite side is across from the given angle, and the adjacent side is net to the given angle, ut is not the hypotenuse. a c For : With respect to, a is the opposite side, and is the adjacent side. With respect to, is the opposite side, and a is the adjacent side. c is always the hypotenuse. Tangents 289

4 The Tangent Ratio One of the most commonly used ratios is the tangent ratio. It is areviated tan. The tangent of an angle is the ratio of the length of the leg opposite that angle to the length of the leg adjacent to that angle. a c tan = opposite adjacent = a _ opposite tan = adjacent = a The tangent ratio in trigonometry is related to tangents to circles. Look at circle. It has a radius of 1. _ is tangent to circle at point. The tangent of is the ratio of the length of the tangent segment to the radius. 1 _ opposite tan = adjacent = _ = _ 1 = That is, the tangent of the angle is the length of the tangent segment. Trigonometric ratios can e epressed as either a fraction or decimal. E F 5 D tan D = _ 12 5 = 2.4 tan E = _ You can use a calculator to find a decimal approimation for a tangent ratio. To do so, set your calculator in degree mode and use the tan key. The screen elow shows that the tangent of a 55 angle is approimately To find the value of in the diagram elow, we set up the tangent ratio and solve for tan 43 = _ (tan 43 ) = Unit 10 TRIGOnometry

5 The Inverse Tangent Sometimes you know the lengths of the legs of a right triangle and you want to find the measure of an acute angle. For this, you use the inverse tangent, areviated tan 1. Your calculator may have tan 1, or you may have to use an INV key. s shown elow, if the tangent of an angle is 5 3, or 0.6, the angle measure is approimately 31. The grade of a hill is given to e 15%. To find the angle of elevation, draw a right triangle with a slope of Then find the inverse tangent tan = tan = The angle of elevation is aout 8.5. Summary The tangent of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle: tan = _ opposite adjacent. You use the tangent of an angle to find the length of a missing leg in a right triangle. To find a missing angle, use the inverse tangent: tan 1. Tangents 291

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7 Sines and osines Ojectives Epress the sine and cosine ratios of given angles as fractions or decimals. Use a calculator to find the value of the sine or cosine of an angle. Use the inverse sine and cosine to find the measure of an angle. Trigonometric ratios can e used to make indirect measurements. For instance, suppose you are flying a kite and let it out to its full length. y knowing the length of the string and y estimating the angle the string is making with the ground, you can estimate how far directly aove the ground the kite is flying. KEYWORDS cosine identity sine The Sine and osine Ratios The tangent ratio only involves the legs of a right triangle, ecause opposite and adjacent (the two legs) are never the hypotenuse. Now you will learn aout two more trigonometric ratios, oth of which involve the hypotenuse. They are sine and cosine, areviated sin and cos. The sine of an angle is the ratio of the length of the leg that is opposite the angle to the length of the hypotenuse. a c opposite sin = hypotenuse = a c sin = opposite hypotenuse = c Sines and osines 293

8 The cosine of an angle is the ratio of the length of the leg adjacent to that angle to the length of the hypotenuse. a c adjacent cos = hypotenuse = c adjacent cos = hypotenuse = a c D 1 E F D 1 sin tan cos E F s with tangent, oth sine and cosine can e defined as lengths of segments on a circle with a radius of 1. Return to circle where _ is a tangent segment,, D, and are collinear, and, E, and are also collinear. _ DE is a half-chord ecause _ is a perpendicular isector of chord _ DF. The original word for half-chord in Sanskrit was translated to raic, and then mistranslated to the Latin word sinus, which we now call sine. opposite sin = hypotenuse = _ DE D = _ DE 1 = DE adjacent cos = hypotenuse = _ E D = _ E 1 = E The illustration at left summarizes how the ratios of sine, cosine, and tangent relate to the circle. lso like the tangent ratio, the sine and cosine ratios can e epressed as either fractions or decimals. E 8 10 F 6 D sin D = _ 8 10 = 0.8 sin E = _ 6 10 = 0.6 _ 6 cos D = 10 = 0.6 cos E = _ 8 10 = 0.8 Notice that the sine of one acute angle is the same as the cosine of the other acute angle from the same triangle. s you did for the tangent ratio, you can use your calculator to find decimal approimations for the sine and cosine of an angle. Use the sin and cos keys on your calculator. The screen elow shows that the sine of a 35 angle is approimately and the cosine is approimately Rememer that your calculator must e set in degree mode. 294 Unit 10 TRIGOnometry

9 Here is an eample of how to use sine to find a missing side length _ sin 61 = (sin 61 ) = nd here is an eample that uses cosine y _ 10 cos 36 = y y(cos 36 ) = 10 _ 10 y = cos 36 y You can use trigonometry to make indirect measurements. Suppose a kite is at the end of a 250 ft string. If the string makes an angle of aout 50 with the ground, we can approimate the height of the kite aove the ground y setting up and solving the following equation sin 50 = (sin 50 ) = The kite is aout 192 feet high. The Inverse Sine and osine To find the measure of an acute angle, given the length of a leg and the length of the hypotenuse, you use the inverse sine and cosine: sin 1 and cos 1. s shown elow, if the sine of an angle is 1 2, or 0.5, the angle measure is 30. If the cosine of an angle is 0.5, the angle measure is 60. Sines and osines 295

10 16-foot ladder is leaning against a house. The top of the ladder reaches 10 feet aove the ground. To find the angle the ase of the ladder makes with the ground, we set up an equation and find the inverse sine _ 10 sin = 16 sin = Trigonometric Identities trigonometric identity is an equation containing a trigonometric ratio that is true for all values of the variale. There are many trigonometric identities. We will eamine two of them and show they are true. _ sin Identity 1 tan = cos MemoryTip To rememer that sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent, use each first letter to make the word sohcahtoa, pronounced soak-a-toe-ah. To show that this is true, we sustitute the ratios for sine and cosine into the equation. opposite hypotenuse tan = adjacent hypotenuse opposite tan = hypotenuse hypotenuse adjacent tan = _ opposite adjacent The last equation is the definition of the tangent ratio, so the original equation is true. Identity 2 sin 2 + cos 2 = 1 y (, y) We can use a circle on the coordinate plane to show that this identity is true. Its center is located at (0, 0) and it has a radius of 1. Since all radii of a circle are congruent, the hypotenuse of the triangle has a radius of 1, and we can write the sine and cosine of angle a as follows: a y (1, 0) y sin a = 1 = y and cos a = 1 = The equation of a circle with its center at the origin and a radius 1 is: ( 0) 2 + (y 0) 2 = y 2 = 1 Now we use the ommutative Property of ddition and sustitute sin a and cos a into the equation for and y. (sin a) 2 + (cos a) 2 = Unit 10 TRIGOnometry

11 To make things a it easier to read, mathematicians usually replace (sin a) 2 with sin 2 a, so the identity can e written in the form sin 2 a + cos 2 a = 1 Summary The sine of an angle is the ratio of the length of the side opposite the angle opposite to the length of the hypotenuse: sin = hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent the adjacent angle to the length of the hypotenuse: cos = hypotenuse. You can use the sine and cosine of an angle to find a missing side of a right triangle. To find a missing angle, you can use the inverses: sin 1 and cos 1. trigonometric identity is an equation containing a trigonometric ratio that is true for all values of the variale. Two trigonometric identities are: tan = _ cos sin and sin2 + cos 2 = 1. Sines and osines 297

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13 Special Right Triangles Ojectives Use trigonometric definitions and special right triangles to derive trigonometric ratios for 30, 45, and 60 angles. Solve prolems involving special right triangles y using trigonometric ratios. In life, there are often many ways to solve a prolem. It s the same with mathematics. For some triangles, you can find a missing side y applying the Pythagorean Theorem, y using a formula, or y writing a proportion. They all give the same answer. So, how do you know which way to go? The choice is yours. The decision may depend on whether or not you have a calculator. Or you can solve a prolem one way, and use another way to check your answer. The Triangle ecause of the measures of its angles, the triangle is commonly used, so knowing the trigonometric ratios for a 45 angle can come in handy. Let us first review the relationship etween the legs and hypotenuse of a triangle. When you construct the diagonal of a square, you create two triangles with a side length of s. You can then use the Pythagorean Theorem to see that the length of the diagonal is 2 times the length of a side. d = s 2 45 s 45 s s 2 + s 2 = d 2 2s 2 = d 2 2s 2 = d 2 s 2 = d Rememer radical epression is not considered simplified when the square root is in the denominator. t right, we rationalized the denominator y multiplying _ 1 y _ Now use s and s 2 to find the sine, cosine, and tangent of a 45 angle. You can use either of the 45 angles. opposite sin 45 = hypotenuse = s _ = 1 = _ 2 s adjacent cos 45 = hypotenuse = s _ = 1 = _ 2 s _ opposite tan 45 = adjacent = s_ s = 1 SPEIL RIGHT TRINGLES 299

14 The Triangle The triangle is also a commonly used triangle, so it is a good idea to know the trigonometric ratios for 30 and 60. When you construct the altitude of an equilateral triangle, you create two triangles with a hypotenuse of s and a ase (the shorter side) of s 2. gain, you can use the Pythagorean Theorem to see that the height is 3 times the length of the shorter side. _ h = s s 60 s 2 ( s 2 ) 2 + h 2 = s 2 h 2 = s 2 s2 4 h 2 _ = 4s2 4 s 2 4 h 2 _ = 3s2 4 s h = 3 2 s Now use s, 2, and s 3 2 to find the sine, cosine, and tangent of a 30 and 60 angle. s opposite sin 30 = hypotenuse = 2 s = s 2s = 2 1 _ opposite sin 60 = hypotenuse = s 3 _ 2 s = _ s 3 2s = 3 2 _ cos 30 = adjacent hypotenuse = s 3 _ 2 s = _ s 3 2s = _ 3 2 cos 60 = adjacent s hypotenuse = 2 s = s 2s = 2 1 _ tan 30 = opposite s adjacent = 2 _ 2s = = _ 1 = _ 3 s 3 2 2s _ tan 60 = opposite adjacent = s 3 2 s 2 = _ 2s 3 2s = Unit 10 TRIGOnometry

15 Using the Trigonometric Ratios to Solve Prolems construction worker stands on a rooftop 65 feet aove the ground. She throws down a rope, which another worker should anchor to the ground at a 30 angle. To the nearest foot, how far away from the ase of the uilding should the rope e anchored to make the correct angle? Use the tangent of a 30 angle to set up and solve a proportion. _ 65 tan 30 = 65 _ = _ 65 3 = The rope should e anchored 113 feet from the ase of the uilding. display screen is shaped like a square with a diagonal length of 14 centimeters. To the nearest tenth of a centimeter, find the length and width of the screen. Use the sine (or cosine) of a 45 angle to set up and solve a proportion. 14 cm _ sin 45 = 14 _ 2 2 = _ 14 2 = 14 2 = The screen has a length and width of aout 9.9 centimeters. SPEIL RIGHT TRINGLES 301

16 Summary You can find the sine, cosine, and tangent of a 45 angle y applying the trigonometric definitions to the sides of a triangle. You can find the sine, cosine, and tangent of a 30 angle and a 60 angle y applying the trigonometric definitions to the sides of a triangle. The trigonometric ratios that can e derived from special right triangles are summarized in the tale. ngle Measure sine cosine tangent _ 3 2 _ _ 2 _ _ You can find the lengths of missing sides in special right triangles y using trigonometric ratios. 302 Unit 10 TRIGOnometry

17 The Laws of Sines and osines Ojectives Recognize when to use the Law of Sines or the Law of osines. Use the Laws of Sines and osines to find missing measures in triangles. The Pythagorean Theorem is an invaluale tool, ut it has its limitations. It is only true for right triangles, and there are many situations that involve triangles that are not right triangles. For eample, imagine that two oats leave a dock at the same time and head in different directions. t any point in time, the two oats and dock make up the vertices of a triangle. The new formulas you will learn will allow you to find side and angle measures for any type of triangle, not just a right triangle. KEYWORD solve a triangle The Law of Sines So far we have used trigonometric ratios only to solve right triangles. We can also use trigonometry to solve triangles that are not right. One way is to use the Law of Sines. We will prove it efore we state it. onsider elow. It is not a right triangle, ut y drawing the altitude _ D, we have created two right triangles: D and D. h a D h First we find the sine of : sin = and solve for h: h = sin Then we find the sine of : h sin = a and solve for h: Now we have two epressions equal to h. y sustitution: Finally, we divide oth sides y a: h = a sin sin = a sin _ sin a = _ sin The Laws of Sines and osines 303

18 y drawing different altitudes, we can show that _ sin a = _ sin = _ sin c. This is the Law of Sines. The Law of Sines c a For any, where a,, and c are the measures of the opposite sides of,, and, respectively, _ sin a = _ sin = _ sin c Using the Law of Sines, you can find a missing triangle measure in the following two cases. You are given two angles and any side (S or S). You are given two sides and the nonincluded angle (SS). When you use the Law of Sines, you set up and solve a proportion using two of the three ratios. To find the length of side a elow, we set up a proportion using the ratios with and. Rememer The largest angle is across from the largest side and the smallest angle is across from the smallest side. For the prolem at right, is the greatest angle, so a must e less than a _ sin a = _ sin sin 55 a = sin a(sin 75 ) = 20(sin 55 ) 20(sin 55 ) a = sin 75 a 17 You can also use the Law of Sines to find angle measures of triangles. To find the measure of E elow, we set up a proportion and use the inverse sine. F 7 10 E 44 D _ sin E e = _ sin D d _ sin E 10 = sin (sin E) = 10(sin 44 ) 10(sin 44 ) sin E = 7 10(sin 44 ) m E = sin ( 1 7 ) Unit 10 TRIGOnometry

19 The Law of osines The Law of Sines can only e used for certain angle and side cominations. When you cannot use the Law of Sines, you may e ale to use the Law of osines. We will prove the law efore we state it. gain, we have with altitude _ D creating two right triangles: D and D. Let D = and D = c. D h c a c First we find the cosine of angle : cos = and we solve for : = cos We use the Pythagorean Theorem for D: 2 = h and solve for h 2 : h 2 = 2 2 Then we use the Pythagorean Theorem in D: a 2 = h 2 + (c ) 2 and square the inomial: a 2 = h 2 + c 2 2c + 2 Now we sustitute for h 2 : a 2 = c 2 2c + 2 comine like terms: a 2 = 2 + c 2 2c sustitute for : a 2 = 2 + c 2 2c( cos ) and rearrange factors: a 2 = 2 + c 2 2c cos We can draw different altitudes and follow similar reasoning for each. The Law of osines c a For any, where a,, and c are the measures of the opposite sides of,, and, respectively, a 2 = 2 + c 2 2c cos 2 = a 2 + c 2 2ac cos c 2 = a a cos Use the Law of osines to find a missing triangle measure in the following two cases. You are given two sides and the included angle (SS). You are given three sides (SSS). When you use the Law of osines to find a missing side, choose the formula that has that side on the left side of the equation. To find the length of side elow, we choose the formula with 2 on the left side = a 2 + c 2 2ac cos 2 = (4)(10) cos 34 2 = cos The Laws of Sines and osines 305

20 If you know the lengths of three sides of a triangle, you can use the Law of osines to find the measure of any angle of the triangle. Two ships leave a dock at the same time and head in different directions without changing course. fter a few hours, one ship is 200 miles from the dock and the other is 325 miles from the dock. They are 410 miles from each other. What angle is formed y the paths of the two ships at the dock? In the diagram, is the missing angle, so we use the formula that has on the right side. Notice elow that we isolate cos and then use the inverse cosine. 410 mi 325 mi 200 mi a 2 = 2 + c 2 2c cos = (200)(325)cos 168,100 = 145, ,000 cos 22,475 = 130,000 cos _ 22, ,000 = cos m 100 If you want to solve the triangle, which means to find all missing measures, you can now use the Law of Sines to find another angle, and then sutract the sum of the two known measures from 180 to find the last angle measure. Summary The Laws of Sines and osines can e used to solve triangles other than right triangles. The Law of Sines states that for any, _ sin a = _ sin = _ sin c. Use the Law of Sines when you are given two angles and any side, or two sides and a nonincluded angle. The Law of osines states that for any, a 2 = 2 + c 2 2c cos 2 = a 2 + c 2 2ac cos c 2 = a a cos Use the Law of osines when you are given two sides and the included angle, or all three sides. 306 Unit 10 TRIGOnometry

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