Intermediate Algebra with Trigonometry. J. Avery 4/99 (last revised 11/03)

Transcription

1 Intermediate lgebra with Trigonometry J. very 4/99 (last revised 11/0)

2 TOPIC PGE TRIGONOMETRIC FUNCTIONS OF CUTE NGLES SPECIL TRINGLES FINDING TRIGONOMETRIC FUNCTIONS USING CLCULTOR SOLVING RIGHT TRINGLES TRIGONOMETRIC PPLICTIONS

3 TRIGONOMETRIC FUNCTIONS OF CUTE NGLES In right triangle BC, the sides are named using the lower-case letters corresponding to the angles opposite the sides, as shown below. Sides a and b are called legs; c is the hypotenuse. (Recall the Pythagorean Theorem, a + b = c.) b c C a B Note that leg a is opposite angle, leg b is opposite angle B and the hypotenuse c is opposite angle C. Right triangles should always be labeled using this pattern. The definitions of the three basic trigonometric functions based on acute angles and B in right BC are defined as follows: Sine (abbreviated as sin ) = Cosine (abbreviated as cos ) = Tangent (abbreviated as tan ) = opposite leg hypotenuse = a c adjacent leg hypotenuse = b c opposite leg adjacent leg = a b Using these basic definitions, sin B = b c, cos B = a c and tan B = b a. Note that these are ratios of the sides of the right triangle, not the angle degree measure. It is often helpful to use an acronym of the first initial of the trigonometric function and Opposite leg their ratios to remember the definitions. Using Sin of the angle =, Cos of Hypotneuse djacent leg Opposite leg the angle = and Tan of the angle =, an acronym such as Hypotneuse djacent leg Some Old Horse Caught nother Horse Taking Oats way can be used where the first letter in each word stands for the trigonometric function and its ratio. Some people find that just remembering the spelling SOHCHTO is helpful.

4 It is important to note that trigonometric ratios are exact values when written as fractions. If an answer contains a radical, it should be in its simplified form. Write answers as rounded decimals only when directed to round an answer. Examples: 1. Find the exact value of the three trigonometric ratios for angles and B if a = 1, b = 5 and c = 1. Solution: It is helpful to sketch a right triangle and label the sides and angles. Remember, side a is opposite angle, side b is opposite angle B, side c is opposite angle C. B sin = opposite leg hypotenuse = cos = adjacent leg hypotenuse = 5 1 C 5 tan = opposite leg adjacent leg = 1 5 sin B = 5 1 cos B = 1 1 tan B = 5 1. Find the exact value of the three trigonometric ratios for angles and B if b = 8 and c = 10. Solution: Use the Pythagorean Theorem to find leg a. a + b = c a + (8) = (10) a + 64 = 100 a = a = 6 C 6 B sin = 6 10 = 5 sin B = 8 10 = 4 5 cos = 8 10 = 4 5 cos B = 6 10 = 5 tan = 6 8 = 4 tan B = 8 6 = 4

5 . Find the exact value of the three trigonometric ratios for angles and B if a = and b =. Leave answers in simplified radical form. Solution: a + b = c () + () = c = c 18 = c = c B C sin = cos = = 6 = = sin B = cos B = = = tan = = 1 tan B = = 1 4. If sin = 4, find cos and tan. Leave answers in simplified radical form. opposite leg Solution: Since sin = hypotenuse, find the adjacent leg for the cosine and tangent ratios. a + b = c () + b = (4) b = 16 b b = 7 b = 7 C B Then, cos = 7 4 and tan = = Given tan B = 4, find sin B and cos B. Leave answers in simplified radical form. opposite leg Solution: Since tan B = adjacent leg, and tan B = 4 = 4 1, then the opposite leg = 4 and the adjacent leg = 1. Using the Pythagorean Theorem, the hypotenuse is found to be 4 17 Then, sin B = 17 = and cos B = 17 =

6 Practice Problems: #1 - : Find the exact value of the three trigonometric ratios for angles and B in right BC using the information given. Leave answers in simplified radical form. 1. a = 1, b =, c = 5. a =, c = 5. a =, b = 5 4. If cos = 7, find sin and tan. 5. If tan B = 1, find sin B and cos B. 5

7 SPECIL TRINGLES There are two right triangles with special characteristics. One is the isosceles right triangle. Recall that an isosceles triangle is one in which two sides are equal in length and the angles opposite those sides are equal in measure. In an isosceles right triangle, the two equal angles each measure 45 and the legs are equal in length. To find the ratio of the sides in a triangle, consider the following examples. Examples: 1. In right BC, a = b =. The length of the hypotenuse is (as found in the previous example). Then sin 45 = = 6 =, cos 45 = and tan 45 = = 1. = 6 =, Note that the measure of the angles in BC were known to be 45 since the legs of the triangle are equal in length.. In BC where a and b are 4 inches long, the hypotenuse c is found to be 4 inches. 4 Then sin 45 = 4 = 4 8 =, cos 45 = and tan 45 = 4 4 = = 4 8 =, It follows that regardless of the length of the legs in an isosceles right triangle, the three trigonometric ratios will always reduce to sin 45 = cos 45 = 1 tan 45 = 1 C 1 B The drawing at the right above shows the lengths of the legs and hypotenuse of a triangle in their lowest terms. 6

8 nother right triangle with special characteristics is the triangle. The leg opposite the 0 angle is exactly half the length of the hypotenuse. Thus in BC where = 0, if leg a =, the hypotenuse c = 4; if a = 7, c = 14. If c = 10, a = 5; if c =, a = 1. Examples: 1. Find the three trigonometric ratios of the 0 and 60 angles in right BC if c = 6 and B is 0. Solution: Since the leg opposite the 0 angle is half the length of the hypotenuse, b =. Find a by the Pythagorean Theorem. a + b = c a + () = (6) 6 a + 9 = 6 a = 7 0 a = B a C sin 0 = 6 = 1 sin 60 = 6 = cos 0 = 6 = cos 60 = 6 = 1 tan 0 = = 9 = tan 60 = =. Given = 0 and a = 5 cm. Find the three trigonometric functions of and B in right BC. Solution: c = 10 cm since the hypotenuse is twice the length of the leg opposite the 0 angle. Find b using the Pythagorean Theorem. B a + b = c (5) + b = (10) 5 + b = b = 75 b = 5 0 C b sin 0 = 5 10 = 1 cos 0 = 5 10 = 5 tan 0 = 5 = 5 15 = sin 60 = 5 10 = cos 60 = 5 10 = 1 tan 60 = 5 5 = 7

9 It follows that regardless of the length of the legs in a triangle, the three trigonometric ratios for the acute angles will always reduce to sin 0 = 1 cos 0 = tan 0 = sin 60 = cos 60 = 1 tan 60 = 0 60 C 1 B The drawing above shows the lengths of the legs and hypotenuse of a triangle in their lowest terms. Here is a summary table that should help you memorize the trigonometric ratios for these special angles. Notice the pattern across the rows for sine and cosine. The tangent value sin can be found by dividing sine by cosine tan = cos sin 1 1 = cos tan = 1 Practice Problems: Find the exact value of the trigonometric ratio. (Do not use a calculator.) 1. sin sin 0. tan 0 6. cos 45. cos tan tan sin 60 8

10 FINDING TRIGONOMETRIC FUNCTIONS USING CLCULTOR calculator is necessary to find the values of a trigonometric functions for angles other than 0, 60 and 45. Since calculators differ in the ways input is accepted, it is important to know how your calculator works. For some calculators, the trigonometric function key,, or is pressed first, then the angle value; SIN COS TN for others, the angle must be entered before the trigonometric function key is pressed. Be sure your calculator is in degree mode. In many calculators, the abbreviation deg will appear on the display if it is in the degree mode. If rad or grad appears instead, press the mode key (or the key that changes modes on your calculator) until deg appears. Try the following examples to determine how your calculator requires input of the trigonometric function and angle. Examples: nswers are rounded to four decimal places. 1. sin 5 = cos 16 = tan 6.45 = When the trigonometric ratio is known and the angle is unknown, the angle can be found using the SIN -1, COS -1, or TN -1 keys. For example, sin = means find the angle whose sine is The solution is found by the equation = sin The calculator key sequence to find the solution is similar to the sequence used to find the SIN trigonometric ratio for an angle. That is, if you used the sequence 5 to get the solution to the first example, you will use SIN to find the measure of angle. If you used the sequence 5 SIN, you will use SIN -1 to find the measure of the angle. Try the following examples on your calculator. 9

11 Examples: nswers are rounded to the nearest degree. 1. sin = = sin = 6. cos = = cos = 9. tan = 1.09 = tan = 87 Practice Problems: Find the trigonometric ratio of the angle rounded to four decimal places. 1. cos 5 =. sin 5 =. sin 10 = 4. tan 89 = 5. cos = 6. tan 40 = Find the measure of the angle to the nearest degree. 7. cos = = 8. sin = = 9. tan = = 10. sin = = 11. tan = 10.1 = 1. cos = 0.0 = 10

12 SOLVING RIGHT TRINGLES Given enough information about the angles and sides of a right triangle, all unknown sides or angles can be found. This is called solving the right triangle. Examples: 1. Given = 6 and c = 96 m in right BC, find B and sides a and b, rounded to the nearest whole meter. Solution: Since the sum of the angles in a triangle is 180 and C is a right angle (90 ), B = 90 - (measure of ) = 90-6 = 54. The triangle in the diagram below shows the angles and the side that are known. Use the trigonometric ratios to find a and b. sin = a c cos = b c 6 C sin 6 = a 96 cos 6 = b 96 a = 96 (sin 6 ) b = 96 (cos 6 ) a = 56 m b = 78 m 96 m B. Given B = 0 and a = 5 ft in right BC, find and sides b and c, rounded to the nearest whole foot. Solution: = 90-0 = 70 tan B = b a cos B = a c tan 0 = b 5 cos 0 = 5 c 0 b = 5 (tan 0 ) c (cos 0 ) = 5 C 5 B b = 19 ft 5 c = cos 0 c = 55 ft Remember, it is often helpful to draw a right triangle and label the sides and angles that are given. This way you should be able to quickly determine which trigonometric functions can be used to solve for the unknown sides. Choose the functions that use the given information, not the answers you have found so answers that are rounded will not affect further results. 11

13 Practice Problems: Solve the right triangle. 1. In right BC, B = 6 and a = 5. Find and sides b and c. Round answers to the nearest tenth.. In right BC, = 15 and a = 10. Find B and sides b and c. Round answers to the nearest tenth. 1

14 TRIGONOMETRIC PPLICTIONS n angle of elevation starts from the horizontal and goes upward. It is formed between the horizontal and the line of sight when looking up at an object. Draw a horizontal ray and a ray going upward. The angle measured between these two rays is the angle of elevation. angle of elevation horizontal angle of elevation horizontal n angle of depression starts from the horizontal and goes downward. It is formed between the horizontal and the line of sight when looking down at an object. Draw a horizontal ray and a ray going downward. The angle measured between these two rays is the angle of depression. horizontal angle of depression horizontal angle of depression Examples: 1. The Washington monument is 555 feet high. If the top of the monument is 700 feet, diagonally, from a marker on the ground, what is the angle of depression from the top of the monument to the marker? Round to the nearest degree. Solution: Draw a sketch of the problem and identify the given parts of the right triangle. Note that the angle of depression is equal to the angle (marked x) formed between the ground and the diagonal. Find x using the sine function. angle of depression sin x = x = sin x x = 5. From a point on the ground, the angle of elevation to the top of a wall is 40. If the wall rises 8.6 meters above the ground, how far is the point on the ground from the bottom of the wall? Round to the nearest tenth. 1

15 Solution: tan 40 = 8.6 x wall x (tan 40 ) = m 8.6 x = tan x = 10. m x. loading platform is 1. meters above the ground. How long must a ramp be in order to make an angle of with the ground? Solution: platform ramp sin = 1. x 1. m x x (sin ) = x = sin x =. m Use the information given to determine which trigonometric function will solve the problem. Labeling a right triangle will help identify the relationship between the known sides and/or angles and the side or angle that is a solution to the application. 14

16 Practice Problems: 1. 5-meter high flagpole casts a shadow 10 meters long. Find the angle of elevation of the sun to the nearest tenth of a degree.. rope is stretched from the top of a vertical pole to a point 6.8 meters from the foot (base) of the pole. The rope makes an angle of 8 with the ground. Find the length of the pole to the nearest tenth of a meter.. From the top of a vertical cliff 400 meters high, the angle of depression to the ground is 18. hang-glider is able to glide from the top of the cliff to the ground in a straight line. What is the distance the hang-glider glides? (Round to the nearest meter.) 15

17 nswers to Practice Problems Page 5 1. sin = 5 5, cos = 5 5, tan = 1, sin B = 5 5, cos B = 5 5, tan B =. sin = 5, cos = 4 5, tan = 4, sin B = 4 5, cos B = 5, tan B = 4. sin = , cos = 9, tan = 5 9 5, sin B = 9, cos B = 9 9, tan B = 5 4. sin = 5 7, tan = sin B = 5, cos B = 5 5 Page Page Page 1 1. b = 9.8, c = 11. b = 7., c = 8.6 Page m. 194 m 16