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### Unit 5 Notes

1. 1. Unit 5Geometry & Trigonometry
2. 2. Geometry & Trigonometry• Geometry is a branch of mathematics that investigates the measurement and relationships of lines, points, and shapes. Trigonometry is a more specific study of the measurement and relationship of sides and angles in a triangle.
3. 3. Geometry & Trigonometry• In this unit you will learn about various properties and relationships in geometry and then apply them as you solve real world problems using geometry and trigonometry. This will involve the calculation of angles and side lengths in various shapes including triangles.
4. 4. Properties of Triangles• You may recall a few types of triangles. A Right Triangle has one angle that is 90°. An Isosceles Triangle has at least two congruent (equal) sides. If there are exactly two congruent sides they are called legs, and its non-congruent side is the base. Angles opposite the equal sides are equal. In an Equilateral Triangle, all three sides are congruent and each of the three angles equal 60°.
5. 5. Properties of Triangles Online Activity
6. 6. Using Trigonometry
7. 7. Using Trigonometry
8. 8. The Sine RatioThe ratio of the length of the side opposite an angle to the length of thehypotenuse is very important in the study of trigonometry. It is referred to as thesine of an angle.
9. 9. The Cosine RatioThe cosine of an angle is the ratio of the length of the side adjacent to a givenangle to the length of the hypotenuse.The word adjacent means close or adjoining. The side adjacent to a particularangle is the side of the triangle remaining after you label the hypotenuse and theside opposite the angle.
10. 10. The Cosine Ratio
11. 11. The Tangent RatioThe tangent of an angle is the ratio of the lengthof the side opposite a given angle to the lengthof the side adjacent to the given angle.
12. 12. The Tangent Ratio
13. 13. Sine, Cosine, and TangentThese three ratios can be shortened to: The symbol θ (theta) is often used in mathematics to refer to an unknown angle.It is a letter of the Greek alphabet. It serves as a reminder of the contributionthat ancient Greece made to the study of mathematics.Some students find it helpful to remember the trigonometric relationships ofsine, cosine, and tangent by the following mnemonic, SOH CAH TOApronounced “sock-a-toe-a”.
14. 14. Sine, Cosine, and Tangent
15. 15. Determining Trig Ratios
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17. 17. Solving Trigonometry Problems – StrategiesTo solve word problems involving trigonometric ratios, note the following 7-stepprocedure: 1. Read the problem carefully. 2. Identify the given and unknown information. 3. Draw a diagram and label it appropriately. 4. Choose and substitute into the appropriate trigonometric ratio. 5. Solve for the unknown. 6. Check your answer to see if it is reasonable. 7. Make a concluding statement.
18. 18. Solving Trigonometry Problems – Strategies
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20. 20. Applications of Pythagorus: CarpentryThe Pythagorean Theorem, a2 + b2 = c2, is used in construction. Carpenters,when raising walls for a house, need to know if the walls are "square" to eachother and form right angles in the corners, before they are pinned down. Oneway they do this is by measuring 3 feet along one wall and making a pencilmark, then measuring 4 feet along the second wall and making another mark.The carpenter will place their tape measure at a diagonal between these twopencil marks and make adjustments to the walls until the tape measure reads 5feet. The carpenter is applying the Pythagorean triple: 32 + 42 = 52.
21. 21. Applications of Trigonometry: NavigationA helicopter is 10 km due west of its base when it receives a call to pick up astranded hiker. The hiker is 15 km due north of the helicopters present position.Once the helicopter picks up the hiker, what is the measure of the anglebetween the route to the base and its current route?Sample Solution:The helicopters route to base is 34 degrees away from its current route.
22. 22. Applications of Trigonometry: ArchitectureA right rectangular pyramid is made of 4 isosceles triangles with a square base.The Great Pyramid in Egypt has a square base with side length 755 ft and aheight of 481 ft. Determine the lateral surface area to the nearest foot.Sample Solution:Calculate the outer triangles slant height by using thePythagorean Theorem.Half of the base is 755 ft/2 = 377.5 ft.Then, calculate the area of this triangle. Lastly, multiply the area by 4 triangles. Area = 230 822 ft2 x 4 Area = 923 289.5 ft2
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25. 25. Properties of Quadrilaterals and Polygons
26. 26. Properties of Quadrilaterals and Polygons
27. 27. Properties of PolygonsA polygon is a figure formed by three or more segments (sides).Convex polygons are polygons in which each interior angle measures lessthan 180°. In other words, the polygon does not "cave" in on any side. InConcave polygons, one or more interior angles may measure more than 180°.Convex: Concave: In a regular polygon, all sides are equal in length and all angles are the same measurement.
28. 28. Interior Angle Properties of PolygonsThe sum of all interior angles in a convex polygon can be found by using theformula:Sum = 180°(n − 2)Where n = the number of sides of the polygon.Example 1:Find the measurement of all the interior angles in a regular polygon with 5 sides. Sample Solution: Sum = 180°(n − 2) and n = 5 Sum = 180°(5 − 2) Sum = 180°(3) Sum = 540°
29. 29. Interior Angle Properties of PolygonsThe sum of all interior angles in a convex polygon can be found by using theformula:Sum = 180°(n − 2) Where n = the number of sides of the polygon.Example 2:Find the measurement of one of the angles in the regular 5-sided polygon.Sample Solution:You found the sum was 540° for all 5 angles, so you divide the sum by 5:
30. 30. Exterior Angle Properties of PolygonsThe sum of all measures of the exterior angles in a regular polygon equals360°. Therefore, to find the measure of one exterior angle, divide by n.
31. 31. Exterior Angle Properties of PolygonsExampleFind the measurement of the exterior angle of a regular 5-sided polygon.Sample Solution:The exterior angle measuresIt is important to note that you just discovered that a 5-sided polygon has interiorangles which each measure 108°.Together, the interior angle (108°) and the exterior angle (72°) add up to 180°.
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34. 34. Law of SinesThe law of sines can be used to calculate the unknown side lengths and angle ina non-right triangle when two angles and a side are known. It can also be usedwhen two sides and one of the angles across from a given side are known. Thelowercase letters represent the sides and the uppercase letters represent theangles opposite those sides. Consider the following diagram:
35. 35. Law of Sines
36. 36. Law of SinesWhen you solve an oblique triangle you cannot use the Pythagorean Theoremor SOH CAH TOA because there is no right angle. One method to solve thesetriangles is to use the law of sines.The law of sines, or the sine law, is an equation that relates the sides of anytriangle to the sine of its angles.According to the law:
37. 37. Using The Law of Sines to find a Missing SideWhen you solve for a missing side in an oblique triangle, you should set up theratio with the sides on top of the law of sines formula. Then, when you solve forthe unknown side, it will be on top making the ratio easier to solve.Example:Calculate the length of side b in the following triangle.
38. 38. Using The Law of Sines to find a Missing SideSample Solution:If you were to label this diagram, you could say:Notice that you do not need to label side c or Angle C as they are not needed inthis diagram. Also notice, that the ratios have been written with the sides on thetop. This will make solving for an unknown side easier.Therefore, you will only use the part of the law of sines that uses the a-valuesand the b-values. To solve for b, you will substitute the correct values in for eachvariable, and then solve algebraically for the missing variable. Therefore, side b equals 5.08 cm.
39. 39. Using The Law of Sines to find a Missing AngleExample:Calculate Angle C in the following triangle.
40. 40. Using The Law of Sines to find a Missing SideSample Solution:The Law of Sines formula is still used, but it is best if you flip the fractions upsidedown when solving for angles. To solve for an angle, substitute the values in foreach variable, solve algebraically, and then take the inverse sine (sin-1) to findthe angle.
41. 41. Applications of the Law of SinesTriangles are thought of as the strongest shape in construction because theyhave the smallest number of sides and angles of any polygon.Carpenters may use the law of sines when calculating the angle of roof peaks,when building trusses to support a roof, or when constructing ramps and slopedwalkways. Calculating the sides and angles of a triangle can then help thecarpenter determine the area needed for surfacing, roofing, or for pouringconcrete.
42. 42. Applications of the Law of SinesExample 1:Dan is building a skateboard ramp. He uses a mitre saw to cut a triangular pieceas a side brace to support the ramp. (A mitre saw can be adjusted to cut atspecific angles within a quarter of a degree).The base of the ramp measures 1.87 metres. For a smooth dismount whenperforming a stunt at the peak of the ramp, the ramp must measure 120° at thepeak.At what angle should Dan cut the wood if the side opposite that angle measures0.74 m?
43. 43. Applications of the Law of SinesSample Solution:Dan should cut the wood at an angle of 20°.
44. 44. Applications of the Law of SinesTrigonometry is applied in navigation, surveying, satellite operations, and navaland aviation industries.Cartographers (map designers) for example and land surveyors need specificcalculations for their profession which may involve the law of sines.These calculations help many industries such as tourism, travel, and aviationwith the distance of trails, heights of mountains, and navigation.
45. 45. Applications of the Law of SinesExample 1:Two hikers want to view some waterfalls. The directwalking distance from their location to the waterfallsis 2.5 kilometres. However, because of thick bushand difficult terrain, the hikers turn at an angle of 22°from their original path, hike for 2 km and then turn135° towards the falls to complete their hike.How much farther did they have to go because oftheir detour?
46. 46. Applications of the Law of SinesSample Solution:To find the total length of the detour, you must find the missing side first. The total length of the detour is 2 km + 1.3 km = 3.3 km. 3.3 km − 2.5 km = 0.8 km. Thus, the hikers travelled 0.8 km farther.
47. 47. Applications of the Law of SinesArchitecture, interior decorating, digital imaging, and music production are allreal life applications that employ trigonometry.Builders of bridges and buildings use angles and trigonometry to engineer astable architectural structure.When redecorating a family room, you may consider the angles and distancebetween lights and speakers of a home theatre system in order to receive thebest sound and brightness for watching a movie. Sound engineers will calculatethese angles for staged concerts to give viewers an optimum experience.
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50. 50. Law of Cosines
51. 51. Law of Cosines
52. 52. Law of CosinesThe law of sines helped you to solve oblique triangles when you knew thevalue of one side, its opposite angle, and one other part of the triangle.However, when you are given an oblique triangle, you are not always givenan opposing pair — a side that matches with an opposite angle.Law of Cosines:  a2 = b2 + c2 − 2bccos(A) There are only two cases when you will have to use the law of cosines: •When you are given all three sides and no angles. •When you are given two sides and the angle between them.
53. 53. Using The Law Of Cosines To Find A Missing SideExample:Calculate side a in the following triangle.
54. 54. Using The Law Of Cosines To Find A Missing SideSample Solution:To solve for side a, substitute in the correct values for each variable, andthen solve algebraically for the missing variable.
55. 55. Using The Law Of Cosines To Find A Missing SideSample Solution:If you want to solve a triangle for side b or side c, you can change theformula to start with b2 = or c2 = and adjust the other variables as long as theformula ends with the same variable that you started with!  a2 = b2 + c2 − 2bccos(A) OR  b2 = a2 + c2 − 2accos(B) OR  c2 = a2 + b2 − 2abcos(C) Please note that when you are solving a question involving the law of cosines you need to be very careful to follow the correct order of operations. Ensure that you find the entire product of 2bc(cosA) before you subtract from the sum of b2 + c2.
56. 56. Using The Law Of Cosines To Find A Missing AngleExample:Determine the measure of ∠C.
57. 57. Using The Law Of Cosines To Find A Missing AngleSample Solution:  You can substitute the known values into c2 = a2 + b2 − 2abcos(C) and thenuse your algebraic skills to rearrange the formula and solve for the angle.However, there are easier versions of the Law of Cosines to solve for anangle.
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