module 7 triangle trigonometry super final

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  • 1. GRADE 9 MATHEMATICS QUARTER 4 Module 7 May 2014

2. Triangle Trigonometry MODULE 7 May 20, 2014 3. In a right triangle, one of the angles measures 90 , and the remaining two angles are acute and complementary. The longest side of a right triangle is known as the hypotenuse and is opposite the right angle. The other two sides are called legs. The leg that is a side of an acute angle is called the side adjacent to the angle. The other leg is the side opposite the angle. Lesson 1 The Six Trigonometric Ratios: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent C B A a c b Hypotenuse Side Opposite Side Adjacent 3 4. Lesson 1 The Six Trigonometric Ratios: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar. If an acute angle of one right triangle is congruent to an acute angle of another right triangle, the triangles are similar, and the ratios of the corresponding sides are equal. Therefore, any two congruent angles of different right triangles will have equal ratios associated with them. The ratios of the sides of the right triangles can be used to define the trigonometric ratios. The ratio of the side opposite and the hypotenuse is known as the sine. The ratio of the side adjacent and the hypotenuse is known as the cosine. The ratio of the side opposite and the side adjacent is known as the tangent. 4 5. SOH-CAH-TOA is a mnemonic device commonly used for remembering these ratios. sin = cos = tan = 5 Lesson 1 The Six Trigonometric Ratios: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent Words Symbol Definition Trigonometric Ratios sine sin sin = cosine cos cos = tangent tan tan = C B A a c b Hypotenuse Side Opposite Side Adjacent 6. 6 Lesson 1 The Six Trigonometric Ratios: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent Words Symbol Definition Reciprocal Trigonometric Ratios cosecant csc = 1 sin secant sec sec = 1 cos cotangent cot cot = 1 tan C B A a c b Hypotenuse Side Opposite Side Adjacent These definitions are called the reciprocal identities. CHO-SHA-CAO is a mnemonic device commonly used for remembering these ratios. csc = sec = cot = In addition to the trigonometric ratios sine, cosine, and tangent, there are three other trigonometric ratios called cosecant, secant, and cotangent. These ratios are the reciprocals of sine, cosine, and tangent, respectively. 7. 7 Lesson 2 Trigonometric Ratios of Special Angles Consider the special relationships among the sides of 30 - 60 - 90 and 45 - 45 - 90 triangles. x x 3 2x 60 30 y y y 2 45 45 These special relationships can be used to determine the trigonometric ratios for 30 , 45 , and 60 . 8. sin cos tan csc sec cot 30 45 60 Activity Complete the table below that summarizes the values of the trigonometric ratios of the angles 30 , 45 , and 60 . 8 Lesson 2 Trigonometric Ratios of Special Angles x x 3 2x 60 30 y y y 2 45 45 9. 9 Answers: Lesson 2 Trigonometric Ratios of Special Angles sin cos tan csc sec cot 30 1 2 3 2 3 3 2 2 3 3 3 45 2 2 2 2 1 2 2 1 60 3 2 1 2 3 2 3 3 2 3 3 10. 10 Lesson 2 Trigonometric Ratios of Special Angles sin cos tan csc sec cot 30 1 2 3 2 3 3 2 2 3 3 3 45 2 2 2 2 1 2 2 1 60 3 2 1 2 3 2 3 3 2 3 3 Notice that sin 30 = cos 60 and cos 30 = sin 60 . This is an example showing that the sine and cosine are cofunctions. That is, if is an acute angle, sin = cos (90 ). Similar relationships hold true for the other trigonometric ratios. 11. 11 Lesson 2 Trigonometric Ratios of Special Angles sin cos tan csc sec cot 30 1 2 3 2 3 3 2 2 3 3 3 45 2 2 2 2 1 2 2 1 60 3 2 1 2 3 2 3 3 2 3 3 sin = cos (90 ) cos = s (90 ) Cofunctions tan = cot (90 ) cot = tan (90 ) sec = csc (90 ) csc = sec (90 ) 12. There are many applications that require trigonometric solutions. For example, surveyors use special instruments to find the measures of angles of elevation and angles of depression. 12 Lesson 3 Angles of Elevation and Angles of Depression 13. 13 Lesson 3 Angles of Elevation and Angles of Depression An angle of elevation is the angle between a horizontal line and the line of sight from an observer to an object at a higher level. 14. 14 Lesson 3 Angles of Elevation and Angles of Depression An angle of depression is the angle between a horizontal line and the line of sight from the observer to an object at a lower level. 15. 15 Lesson 3 Angles of Elevation and Angles of Depression The angle of elevation and the angle of depression are equal in measure because they are alternate interior angles. 16. 16 Lesson 4 Word Problems Involving Right Triangles Trigonometric functions can be used to solve word problems involving right triangles. The most common functions used are the sine, cosine, and tangent. Moreover, you can use trigonometric functions and inverse relations to solve right triangles. To solve a triangle means to find all the measures of its sides and angles. Usually, two measures are given. Then you can find the remaining measures. 17. 17 Lesson 4 Word Problems Involving Right Triangles Example 1. A ladder is 12 feet long. a) If the ladder is placed against a wall so that its base is 2 feet from the wall, find, to the nearest degree, the acute angle the ladder makes with the ground. b) Suppose the base of the ladder is feet from the wall. Find an expression for , the angle the ladder makes with the ground. 18. 18 Lesson 4 Word Problems Involving Right Triangles Example 1. A ladder is 12 feet long. a) If the ladder is placed against a wall so that its base is 2 feet from the wall, find, to the nearest degree, the acute angle the ladder makes with the ground. Solution. cos = 2 12 = 1 1 6 80 19. 19 Lesson 4 Word Problems Involving Right Triangles Example 1. A ladder is 12 feet long. b) Suppose the base of the ladder is feet from the wall. Find an expression for , the angle the ladder makes with the ground. Solution. cos = 12 = 1 12 20. 20 Lesson 4 Word Problems Involving Right Triangles Example 2. Latashi and Markashi are flying kites on a windy day. Latashi has released 250 feet of string, and Markashi has released 225 feet of string. The angle that Latashis kite string makes with the horizontal is 35 . The angle that Markashis kite string makes with the horizontal is 42 . Which kite is higher and by how much? 21. 21 Lesson 4 Word Problems Involving Right Triangles Solution. For Latashis kite: sin 35 = 250 = 250 sin 35 = 143.3941091 Latashis kite has a height about 143.39 ft. 250 ft 35 Height = ? 22. 22 Lesson 4 Word Problems Involving Right Triangles Solution. For Markashis kite: sin 42 = 225 = 225 sin 42 = 150.5543864 Markashis kite has a height about 150.55 ft. 225 ft 42 Height = ? 23. 23 Lesson 4 Word Problems Involving Right Triangles Solution. Lets subtract the height of Markashis kite and the height of Latashis kite. 150.5543864 143.3941091 = 7.160277343 Markashis kite is higher than Latashis kite by about 7.16 ft. 24. 24 Lesson 5 Oblique Triangles Trigonometry enables sides and angle measures to be found in triangles other than right triangles. An oblique triangle is one that does not contain a right angle. Oblique triangles may be classified into two---acute and obtuse. An acute triangle is one that has three acute angles. An obtuse triangle is one that has one obtuse angle. 25. 25 Lesson 5 Oblique Triangles Activity Identify the acute and obtuse triangles. 26. 26 Lesson 5 Oblique Triangles Activity Identify the acute and obtuse triangles. 27. 27 Lesson 5.1 The Law of Sines and Its Applications Law of Sines Let be any triangle with , , and representing the measures of the sides opposite the angles with measures , , and , respectively. Then, the following are true. sin = sin = sin sin = sin = sin 28. 28 Lesson 5.1 The Law of Sines and Its Applications From geometry, you know that a unique triangle can be formed if you know a) the measures of two angles and the included side (ASA) or b) the measures of two angles and the non-included side (AAS). Therefore, there is one unique solution when you use the Law of Sines to solve a triangle given the measures two angles and one side. 29. 29 Lesson 5.1 The Law of Sines and Its Applications From geometry, you know that c) the measures of two sides and a non-included angle (SSA) do not necessarily define a unique triangle. However, one of the following will be true. 1. No triangle exists. 2. Exactly one triangle exists. 3. Two triangles exist. In other words, there may be no solution, one solution, or two solutions. A situation with two solutions is called the ambiguous case. 30. 30 Lesson 5.1 The Law of Sines and Its Applications Suppose you know the measures , , and . Consider the following cases. 31. 31 Lesson 5.1 The Law of Sines and Its Applications ASA Case Example 1. Cartography To draw a map, a cartographer needed to find the distances between point across the lake and each of point and on another side. The cartographer found 0.3 miles, 50 , and 100 . Find the distances from to and from to . 32. 32 Lesson 5.1 The Law of Sines and Its Applications ASA Example 1 Solution. 180 100 50 30 sin 100 = sin 30 0.3 sin 30 = 0.3 sin 100 = 0.3 sin 100 sin 30 0.59 The distance from to is about 0.59 miles. 0.3 mi 50 100 33. 33 Lesson 5.1 The Law of Sines and Its Applications ASA Example 1 (Continuation) Solution. 180 100 50 30 sin 50 = sin 30 0.3 Y sin 30 = 0.3 sin 50 Y = 0.3 sin 50 sin 30 Y 0.46 The distance from to is about 0.46 miles. 0.3 mi 50 100 34. 34 Lesson 5.1 The Law of Sines and Its Applications AAS Case Example 2. A hill slopes upward at an angle of 5 with the horizontal. A tree grows vertically on the hill. When the

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