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Textbook: Trigonometry
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Page 1 chaPTer 6 TRIGONOMETRY 149 6 Trigonometry This chapter is about a very important group of functions called trigonometric functions. The part of mathematics that has to do with these functions is called trigonometry. • You may have applied trigonometric functions in your technical subjects. If this is so, you know what these functions do. However, you may be uncertain about how and why they do it. Anything you did not understand about them will become clear as you work through this chapter. It is very important to go back to those places where you used trigonometry to make sure that you understand how and why the functions worked. In this chapter, you will learn that: • the corresponding sides of similar right-angled triangles are in the same ratio (NB: this is generally true for all similar polygons) • in right-angled triangles, trigonometric functions link input acute angles to output length ratios of the sides of the triangle • in the Cartesian plane, trigonometric functions link input angles of all sizes to ratios, also called quotients, involving coordinates In this chapter, you will learn: • to determine input and output values of the trigonometric functions using diagrams, measurements, and calculations • to use trigonometry to solve equations and to solve problems that have to do with angles, and lengths or coordinates • to use graphs, diagrams, and algebra to represent trigonometric functions Keep in mind that understanding trigonometry and being able to use it effectively as a tool will empower you as a technical person. 2153 TechMaths Eng G10 LB.indb 149 2015/10/22 3:40 PM Page 2 chaPTer 6 TRIGONOMETRY 149 6 Trigonometry This chapter is about a very important group of functions called trigonometric functions. The part of mathematics that has to do with these functions is called trigonometry. • You may have applied trigonometric functions in your technical subjects. If this is so, you know what these functions do. However, you may be uncertain about how and why they do it. Anything you did not understand about them will become clear as you work through this chapter. It is very important to go back to those places where you used trigonometry to make sure that you understand how and why the functions worked. In this chapter, you will learn that: • the corresponding sides of similar right-angled triangles are in the same ratio (NB: this is generally true for all similar polygons) • in right-angled triangles, trigonometric functions link input acute angles to output length ratios of the sides of the triangle • in the Cartesian plane, trigonometric functions link input angles of all sizes to ratios, also called quotients, involving coordinates In this chapter, you will learn: • to determine input and output values of the trigonometric functions using diagrams, measurements, and calculations • to use trigonometry to solve equations and to solve problems that have to do with angles, and lengths or coordinates • to use graphs, diagrams, and algebra to represent trigonometric functions Keep in mind that understanding trigonometry and being able to use it effectively as a tool will empower you as a technical person. 2153 TechMaths Eng G10 LB.indb 149 2015/10/22 3:40 PM 150 Technical Ma TheMaTicS Grade 10 6.1 introducing the roof pitch situation r oof pitch low pitch medium pitch high pitch run rise pitch angle The pitch of a roof tells you how steep the roof is. Pitch can be measured in two ways as: • an angle, measured in degrees or radians, or • a slope, a dimensionless ratio of vertical rises to horizontal run. An important mathematical question: How can we change between the pitch as an angle and the pitch as a rise : run ratio? We will address the question posed above as we work through the chapter. Along the way, we will learn some new math. 2153 TechMaths Eng G10 LB.indb 150 2015/10/22 3:40 PM Page 3 chaPTer 6 TRIGONOMETRY 149 6 Trigonometry This chapter is about a very important group of functions called trigonometric functions. The part of mathematics that has to do with these functions is called trigonometry. • You may have applied trigonometric functions in your technical subjects. If this is so, you know what these functions do. However, you may be uncertain about how and why they do it. Anything you did not understand about them will become clear as you work through this chapter. It is very important to go back to those places where you used trigonometry to make sure that you understand how and why the functions worked. In this chapter, you will learn that: • the corresponding sides of similar right-angled triangles are in the same ratio (NB: this is generally true for all similar polygons) • in right-angled triangles, trigonometric functions link input acute angles to output length ratios of the sides of the triangle • in the Cartesian plane, trigonometric functions link input angles of all sizes to ratios, also called quotients, involving coordinates In this chapter, you will learn: • to determine input and output values of the trigonometric functions using diagrams, measurements, and calculations • to use trigonometry to solve equations and to solve problems that have to do with angles, and lengths or coordinates • to use graphs, diagrams, and algebra to represent trigonometric functions Keep in mind that understanding trigonometry and being able to use it effectively as a tool will empower you as a technical person. 2153 TechMaths Eng G10 LB.indb 149 2015/10/22 3:40 PM 150 Technical Ma TheMaTicS Grade 10 6.1 introducing the roof pitch situation r oof pitch low pitch medium pitch high pitch run rise pitch angle The pitch of a roof tells you how steep the roof is. Pitch can be measured in two ways as: • an angle, measured in degrees or radians, or • a slope, a dimensionless ratio of vertical rises to horizontal run. An important mathematical question: How can we change between the pitch as an angle and the pitch as a rise : run ratio? We will address the question posed above as we work through the chapter. Along the way, we will learn some new math. 2153 TechMaths Eng G10 LB.indb 150 2015/10/22 3:40 PM chaPTer 6 TRIGONOMETRY 151 6.2 revision of angles, lengths, length ratios, and similarity comparing angles and lengths example angles and lengths Imagine Muvhuso is standing at position O on a football field: M P Q R S O A’s path D’s path B’s path C’s path Four of her friends, A, B, C, and D, move from position M along the paths shown. Check that you agree with the following as seen from her position at O: • C ends up furthest from her starting position at M. • B has covered the shortest distance, i.e. B has the smallest path length. • A and B have moved through the same angle. The directions of their starting positions are the same and the directions of their end positions are the same as seen from her position at O. • The change in the direction of the position of C is 90° as seen from her position at O. • D’s path is definitely the longest. But the direction of his end position as seen from her position at O, is the same as the direction of his starting position so the angle change is 0° as seen from her position at O. 2153 TechMaths Eng G10 LB.indb 151 2015/10/22 3:40 PM Page 4 chaPTer 6 TRIGONOMETRY 149 6 Trigonometry This chapter is about a very important group of functions called trigonometric functions. The part of mathematics that has to do with these functions is called trigonometry. • You may have applied trigonometric functions in your technical subjects. If this is so, you know what these functions do. However, you may be uncertain about how and why they do it. Anything you did not understand about them will become clear as you work through this chapter. It is very important to go back to those places where you used trigonometry to make sure that you understand how and why the functions worked. In this chapter, you will learn that: • the corresponding sides of similar right-angled triangles are in the same ratio (NB: this is generally true for all similar polygons) • in right-angled triangles, trigonometric functions link input acute angles to output length ratios of the sides of the triangle • in the Cartesian plane, trigonometric functions link input angles of all sizes to ratios, also called quotients, involving coordinates In this chapter, you will learn: • to determine input and output values of the trigonometric functions using diagrams, measurements, and calculations • to use trigonometry to solve equations and to solve problems that have to do with angles, and lengths or coordinates • to use graphs, diagrams, and algebra to represent trigonometric functions Keep in mind that understanding trigonometry and being able to use it effectively as a tool will empower you as a technical person. 2153 TechMaths Eng G10 LB.indb 149 2015/10/22 3:40 PM 150 Technical Ma TheMaTicS Grade 10 6.1 introducing the roof pitch situation r oof pitch low pitch medium pitch high pitch run rise pitch angle The pitch of a roof tells you how steep the roof is. Pitch can be measured in two ways as: • an angle, measured in degrees or radians, or • a slope, a dimensionless ratio of vertical rises to horizontal run. An important mathematical question: How can we change between the pitch as an angle and the pitch as a rise : run ratio? We will address the question posed above as we work through the chapter. Along the way, we will learn some new math. 2153 TechMaths Eng G10 LB.indb 150 2015/10/22 3:40 PM chaPTer 6 TRIGONOMETRY 151 6.2 revision of angles, lengths, length ratios, and similarity comparing angles and lengths example angles and lengths Imagine Muvhuso is standing at position O on a football field: M P Q R S O A’s path D’s path B’s path C’s path Four of her friends, A, B, C, and D, move from position M along the paths shown. Check that you agree with the following as seen from her position at O: • C ends up furthest from her starting position at M. • B has covered the shortest distance, i.e. B has the smallest path length. • A and B have moved through the same angle. The directions of their starting positions are the same and the directions of their end positions are the same as seen from her position at O. • The change in the direction of the position of C is 90° as seen from her position at O. • D’s path is definitely the longest. But the direction of his end position as seen from her position at O, is the same as the direction of his starting position so the angle change is 0° as seen from her position at O. 2153 TechMaths Eng G10 LB.indb 151 2015/10/22 3:40 PM 152 Technical Ma TheMaTicS Grade 10 length ratios A ratio is one way of comparing two values. Ratios tell us how many times bigger or smaller one measurement is compared to another measurement. Ratios are dimensionless, i.e. they have no unit. In trigonometry, we are particularly interested in length ratios in right-angled triangles. example length ratio Wooden beams: beam P has length a, and beam Q has length b. b = 6,25 m long a = 8,75 m long The ratio of the length of beam P to the length of beam Q can be written in these ways: In ratio notation: a : b = 8,75:6,25 As a whole number ratio: 8,75 ÷ 0,25 = 35 and 6,25 ÷ 0,25 = 25, so a : b = 35:25 As the simplest whole number ratio: 5 is the highest common factor of 35 and 25 so a : b = 7:5. We read this as a is to b as 7 is to 5. In rational form (quotient form): a __ b = 8,75 _____ 6,25 = 7 __ 5 We can change the rational form into decimal form since 7 ÷ 5 = 1,4 then: a __ b = 1,4 Decimal form is very useful. It tells us that beam P is 1,4 times longer than beam Q. We can also write the ratio as a : b = 1,4:1 using the ratio notation. We can also express the reciprocal ratio, the ratio of the length of beam Q to the length of beam P: In ratio notation as b:a = 5:7 = 1:1,4 = 0,71:1 In rational form and in decimal form as b __ a = 6,25 _____ 8,75 = 5 __ 7 . This tells us that the length of beam Q is the fraction 0,71 of the length of beam P. There is another way we can compare two measurements, by finding the difference between their two values. A difference tells us how much bigger or smaller one value is compared to another. Differences have the same unit as the two values. note: Beam P is 2,50 m longer than beam q. This is NOT a ratio statement. This is a difference statement. Beam M of length 7,50 m is also 2,50 m longer than beam N of length 5,0 m. But the ratio of the lengths of beams M and N is 7,5:5 = 3:2 = 1,5:1. So beam M is 1,5 times longer than beam N, which is not the same as for beams P and q, where we have 1,4:1. 2153 TechMaths Eng G10 LB.indb 152 2015/10/22 3:40 PM Page 5 chaPTer 6 TRIGONOMETRY 149 6 Trigonometry This chapter is about a very important group of functions called trigonometric functions. The part of mathematics that has to do with these functions is called trigonometry. • You may have applied trigonometric functions in your technical subjects. If this is so, you know what these functions do. However, you may be uncertain about how and why they do it. Anything you did not understand about them will become clear as you work through this chapter. It is very important to go back to those places where you used trigonometry to make sure that you understand how and why the functions worked. In this chapter, you will learn that: • the corresponding sides of similar right-angled triangles are in the same ratio (NB: this is generally true for all similar polygons) • in right-angled triangles, trigonometric functions link input acute angles to output length ratios of the sides of the triangle • in the Cartesian plane, trigonometric functions link input angles of all sizes to ratios, also called quotients, involving coordinates In this chapter, you will learn: • to determine input and output values of the trigonometric functions using diagrams, measurements, and calculations • to use trigonometry to solve equations and to solve problems that have to do with angles, and lengths or coordinates • to use graphs, diagrams, and algebra to represent trigonometric functions Keep in mind that understanding trigonometry and being able to use it effectively as a tool will empower you as a technical person. 2153 TechMaths Eng G10 LB.indb 149 2015/10/22 3:40 PM 150 Technical Ma TheMaTicS Grade 10 6.1 introducing the roof pitch situation r oof pitch low pitch medium pitch high pitch run rise pitch angle The pitch of a roof tells you how steep the roof is. Pitch can be measured in two ways as: • an angle, measured in degrees or radians, or • a slope, a dimensionless ratio of vertical rises to horizontal run. An important mathematical question: How can we change between the pitch as an angle and the pitch as a rise : run ratio? We will address the question posed above as we work through the chapter. Along the way, we will learn some new math. 2153 TechMaths Eng G10 LB.indb 150 2015/10/22 3:40 PM chaPTer 6 TRIGONOMETRY 151 6.2 revision of angles, lengths, length ratios, and similarity comparing angles and lengths example angles and lengths Imagine Muvhuso is standing at position O on a football field: M P Q R S O A’s path D’s path B’s path C’s path Four of her friends, A, B, C, and D, move from position M along the paths shown. Check that you agree with the following as seen from her position at O: • C ends up furthest from her starting position at M. • B has covered the shortest distance, i.e. B has the smallest path length. • A and B have moved through the same angle. The directions of their starting positions are the same and the directions of their end positions are the same as seen from her position at O. • The change in the direction of the position of C is 90° as seen from her position at O. • D’s path is definitely the longest. But the direction of his end position as seen from her position at O, is the same as the direction of his starting position so the angle change is 0° as seen from her position at O. 2153 TechMaths Eng G10 LB.indb 151 2015/10/22 3:40 PM 152 Technical Ma TheMaTicS Grade 10 length ratios A ratio is one way of comparing two values. Ratios tell us how many times bigger or smaller one measurement is compared to another measurement. Ratios are dimensionless, i.e. they have no unit. In trigonometry, we are particularly interested in length ratios in right-angled triangles. example length ratio Wooden beams: beam P has length a, and beam Q has length b. b = 6,25 m long a = 8,75 m long The ratio of the length of beam P to the length of beam Q can be written in these ways: In ratio notation: a : b = 8,75:6,25 As a whole number ratio: 8,75 ÷ 0,25 = 35 and 6,25 ÷ 0,25 = 25, so a : b = 35:25 As the simplest whole number ratio: 5 is the highest common factor of 35 and 25 so a : b = 7:5. We read this as a is to b as 7 is to 5. In rational form (quotient form): a __ b = 8,75 _____ 6,25 = 7 __ 5 We can change the rational form into decimal form since 7 ÷ 5 = 1,4 then: a __ b = 1,4 Decimal form is very useful. It tells us that beam P is 1,4 times longer than beam Q. We can also write the ratio as a : b = 1,4:1 using the ratio notation. We can also express the reciprocal ratio, the ratio of the length of beam Q to the length of beam P: In ratio notation as b:a = 5:7 = 1:1,4 = 0,71:1 In rational form and in decimal form as b __ a = 6,25 _____ 8,75 = 5 __ 7 . This tells us that the length of beam Q is the fraction 0,71 of the length of beam P. There is another way we can compare two measurements, by finding the difference between their two values. A difference tells us how much bigger or smaller one value is compared to another. Differences have the same unit as the two values. note: Beam P is 2,50 m longer than beam q. This is NOT a ratio statement. This is a difference statement. Beam M of length 7,50 m is also 2,50 m longer than beam N of length 5,0 m. But the ratio of the lengths of beams M and N is 7,5:5 = 3:2 = 1,5:1. So beam M is 1,5 times longer than beam N, which is not the same as for beams P and q, where we have 1,4:1. 2153 TechMaths Eng G10 LB.indb 152 2015/10/22 3:40 PM chaPTer 6 TRIGONOMETRY 153 right-angled triangles: the roof pitch A very important skill you must learn in trigonometry is to see the hidden right-angled triangles in situations. Once you become good at this, things are quite straightforward. We are beginning with the roof pitch situation, the slope or pitch of a roof. Roofers use two ways of describing the pitch (slope) of a roof: • The pitch angle • The rise : run ratio Are these equivalent, that is, the same in some way? Are there different ways of expressing the same thing? Look at the following gable-end of the building in the roof pitch situation: Three yellow triangles have been drawn over the picture. Their hypotenuses lie along the slope of the roof. The remaining sides are horizontal, that is, parallel with the ground, and vertical, that is, straight up from the ground. For a roof pitch the horizontal side is called the run length, vertical side the rise length, and the hypotenuse the rafter length. We can represent these three triangles as projections of each other using a vanishing point, making sure that the corresponding sides are parallel to each other: note that we will use the term ‘rafter length’ to mean the distance between two points along the slope of the roof that corresponds to the particular rise and run we are looking at, and not just to refer to the length of the rafters themselves. 2153 TechMaths Eng G10 LB.indb 153 2015/10/22 3:40 PMRead More
FAQs on Textbook: Trigonometry
| 1. What is the importance of learning trigonometric ratios in Grade 10? |  |
Ans. Learning trigonometric ratios in Grade 10 is crucial because they form the foundation for understanding more complex mathematical concepts. Trigonometric ratios, such as sine, cosine, and tangent, are used to solve problems related to angles and sides in right triangles. This knowledge is not only essential for higher-level mathematics but also has practical applications in fields like physics, engineering, and architecture.
| 2. How do you find the values of sine, cosine, and tangent for common angles? | |
Ans. The values of sine, cosine, and tangent for common angles (0°, 30°, 45°, 60°, and 90°) can be memorized or derived from the unit circle. For example, sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3. Knowing these values helps in quickly solving trigonometric problems without a calculator.
| 3. What are the applications of trigonometry in real life? | |
Ans. Trigonometry has numerous real-life applications, including in fields such as architecture (for designing buildings), engineering (for analyzing forces), astronomy (for calculating distances between celestial bodies), and even in computer graphics (for rendering images). Understanding trigonometry allows us to model and solve various practical problems.
| 4. How can I solve basic trigonometric equations? | |
Ans. To solve basic trigonometric equations, start by isolating the trigonometric function on one side of the equation. Use known values, identities, and the unit circle to find the angle(s) that satisfy the equation. Remember to consider all possible solutions within the specified interval.
| 5. What resources can help me prepare for my Grade 10 trigonometry exam? | |
Ans. To prepare for your Grade 10 trigonometry exam, you can use various resources such as your textbook, online educational platforms (like Khan Academy), video tutorials on YouTube, and practice worksheets. Additionally, joining study groups or seeking help from teachers can provide valuable insights and support.
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