Application of trigonometry

\( \newcommand\): The angle of elevation. Example \(\PageIndex\): The angle of depression Example \(\PageIndex\) Answer: 57,300 feet Problems 1. At a point 60 feet from a tree the angle of elevation of the top of the tree is \(40^\). How high is the mountain (nearest tenth of a mile)?

11.1: Applications of Sinusoids \begin \closegraphsfile 11.2: The Law of Sines \subsection \closegraphsfile 11.3: The Law of Cosines \subsection \closegraphsfile 11.4: Polar Coordinates \subsection \closegraphsfile 11.5: Graphs of Polar Equations \subsection \closegraphsfile 11.6: Hooked on Conics Again \subsection \closegraphsfile 11.7: Polar Form of Complex Numbers \subsection \closegraphsfile 11.8: Vectors \subsection \closegraphsfile 11.9: The Dot Product and Projection \subsection \closegraphsfile 11.10: Parametric Equations \subsection ParseError: EOF expected (click for details) Callstack: at (Under_Construction/Purgatory/Book:_Precalculus_(Stitz-Zeager)/11:_Applications_of_Trigonometry/11.E:_Applications_of_Trigonometry_(Exercises)), /content/body/div[10]/p[272]/span, line 1, column 4 at wiki.page() at (Courses/Chabot_College/MTH_36:_Trigonometry_(Gonzalez)/02:_Applications_of_Trigonometry/2.E:_Applications_of_Trigonometry_(Exercises)), /content/body/div/pre, line 2, column 14 \normalsize \parafcn$ for $t \geq 0$. To find when the hammer hits the ground, we solve $y(t) = 0$ and get $t \approx -0.23$ or $1.61$. Since $t \geq 0$, the hammer hits the ground after approximately $t = 1.61$ seconds after it was launched into the air. To find how far away the hammer hits the ground, we find $x(1.61) \approx 39.

\( \newcommand\) • • • • • • • • • Throughout its early development, trigonometry was often used as a means of indirect measurement, e.g. determining large distances or lengths by using measurements of angles and small, known distances. Today, trigonometry is widely used in physics, astronomy, engineering, navigation, surveying, and various fields of mathematics and other disciplines. In this section we will see some of the ways in which trigonometry can be applied. Your calculator should be in degree mode for these examples. Example 1.11 A person stands \(150\) ft away from a flagpole and measures an angle of elevation of \(32^\circ\) from his horizontal line of sight to the top of the flagpole. Assume that the person's eyes are a vertical distance of 6 ft from the ground. What is the height of the flagpole? Solution: The picture on the right describes the situation. We see that the height of the flagpole is \(h + 6\) ft, where \[\frac\). Example 1.12 A person standing \(400\) ft from the base of a mountain measures the angle of elevation from the ground to the top of the mountain to be \(25^\circ \). The person then walks \(500\) ft straight back and measures the angle of elevation to now be \(20^\circ \). How tall is the mountain? Solution: We will assume that the ground is flat and not inclined relative to the base of the mountain. Let \(h\) be the height of the mountain, and let \(x\) be the distance from the base of the mountain to the point directly beneath the top ...

\( \newcommand\) • In this chapter, we will explore applications of trigonometry that will enable us to solve many different kinds of problems, including finding the height of a tree. We extend topics we introduced in Trigonometric Functions and investigate applications more deeply and meaningfully. • 5.0: Prelude to Applications of Trigonometry The world’s largest tree by volume, named General Sherman, stands 274.9 feet tall and resides in Northern California. Just how do scientists know its true height? A common way to measure the height involves determining the angle of elevation, which is formed by the tree and the ground at a point some distance away from the base of the tree. This method is much more practical than climbing the tree and dropping a very long tape measure. • 5.1: Non-right Triangles - Law of Sines In this section, we will find out how to solve problems involving non-right triangles. The Law of Sines can be used to solve oblique triangles. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side. There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution. • 5.2: Non-right Triangles - Law of Cosines Unfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with triangles where the known angl...

11.1: Applications of Sinusoids \begin \closegraphsfile 11.2: The Law of Sines \subsection \closegraphsfile 11.3: The Law of Cosines \subsection \closegraphsfile 11.4: Polar Coordinates \subsection \closegraphsfile 11.5: Graphs of Polar Equations \subsection \closegraphsfile 11.6: Hooked on Conics Again \subsection \closegraphsfile 11.7: Polar Form of Complex Numbers \subsection \closegraphsfile 11.8: Vectors \subsection \closegraphsfile 11.9: The Dot Product and Projection \subsection \closegraphsfile 11.10: Parametric Equations \subsection ParseError: EOF expected (click for details) Callstack: at (Under_Construction/Purgatory/Book:_Precalculus_(Stitz-Zeager)/11:_Applications_of_Trigonometry/11.E:_Applications_of_Trigonometry_(Exercises)), /content/body/div[10]/p[272]/span, line 1, column 4 at wiki.page() at (Courses/Chabot_College/MTH_36:_Trigonometry_(Gonzalez)/02:_Applications_of_Trigonometry/2.E:_Applications_of_Trigonometry_(Exercises)), /content/body/div/pre, line 2, column 14 \normalsize \parafcn$ for $t \geq 0$. To find when the hammer hits the ground, we solve $y(t) = 0$ and get $t \approx -0.23$ or $1.61$. Since $t \geq 0$, the hammer hits the ground after approximately $t = 1.61$ seconds after it was launched into the air. To find how far away the hammer hits the ground, we find $x(1.61) \approx 39.

\( \newcommand\) • In this chapter, we will explore applications of trigonometry that will enable us to solve many different kinds of problems, including finding the height of a tree. We extend topics we introduced in Trigonometric Functions and investigate applications more deeply and meaningfully. • 5.0: Prelude to Applications of Trigonometry The world’s largest tree by volume, named General Sherman, stands 274.9 feet tall and resides in Northern California. Just how do scientists know its true height? A common way to measure the height involves determining the angle of elevation, which is formed by the tree and the ground at a point some distance away from the base of the tree. This method is much more practical than climbing the tree and dropping a very long tape measure. • 5.1: Non-right Triangles - Law of Sines In this section, we will find out how to solve problems involving non-right triangles. The Law of Sines can be used to solve oblique triangles. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side. There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution. • 5.2: Non-right Triangles - Law of Cosines Unfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with triangles where the known angl...

\( \newcommand\): The angle of elevation. Example \(\PageIndex\): The angle of depression Example \(\PageIndex\) Answer: 57,300 feet Problems 1. At a point 60 feet from a tree the angle of elevation of the top of the tree is \(40^\). How high is the mountain (nearest tenth of a mile)?

\( \newcommand\) • • • • • • • • • Throughout its early development, trigonometry was often used as a means of indirect measurement, e.g. determining large distances or lengths by using measurements of angles and small, known distances. Today, trigonometry is widely used in physics, astronomy, engineering, navigation, surveying, and various fields of mathematics and other disciplines. In this section we will see some of the ways in which trigonometry can be applied. Your calculator should be in degree mode for these examples. Example 1.11 A person stands \(150\) ft away from a flagpole and measures an angle of elevation of \(32^\circ\) from his horizontal line of sight to the top of the flagpole. Assume that the person's eyes are a vertical distance of 6 ft from the ground. What is the height of the flagpole? Solution: The picture on the right describes the situation. We see that the height of the flagpole is \(h + 6\) ft, where \[\frac\). Example 1.12 A person standing \(400\) ft from the base of a mountain measures the angle of elevation from the ground to the top of the mountain to be \(25^\circ \). The person then walks \(500\) ft straight back and measures the angle of elevation to now be \(20^\circ \). How tall is the mountain? Solution: We will assume that the ground is flat and not inclined relative to the base of the mountain. Let \(h\) be the height of the mountain, and let \(x\) be the distance from the base of the mountain to the point directly beneath the top ...

Amongst the lay public of non-mathematicians and non-scientists, Thomas Paine's statement [ ] In Chapter XI of The scientific principles that man employs to obtain the foreknowledge of an eclipse, or of any thing else relating to the motion of the heavenly bodies, are contained chiefly in that part of science that is called trigonometry, or the properties of a triangle, which, when applied to the study of the heavenly bodies, is called astronomy; when applied to direct the course of a ship on the ocean, it is called navigation; when applied to the construction of figures drawn by a ruler and compass, it is called geometry; when applied to the construction of plans of edifices, it is called architecture; when applied to the measurement of any portion of the surface of the earth, it is called land-surveying. In fine, it is the soul of science. It is an eternal truth: it contains the mathematical demonstration of which man speaks, and the extent of its uses are unknown. History [ ] Great Trigonometrical Survey [ ] This section does not Please help Find sources: · · · · ( August 2019) ( Scientific fields that make use of trigonometry include: That these fields involve trigonometry does not mean knowledge of trigonometry is needed in order to learn anything about them. It does mean that some things in these fields cannot be understood without trigonometry. For example, a professor of In some of the fields of endeavor listed above it is easy to imagine how trigonometry could be ...