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Trigonometric function introduction

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sin ( 3 0 ) = 31 , 680 C size 12{"sin" \( 3 rSup { size 8{0} } \) = { {"31","680"} over {C} } } {}

Rearranging terms we find

C = 31 , 680 sin ( 3 o ) size 12{C= { {"31","680"} over {"sin" \( 3 rSup { size 8{o} } \) } } } {}

Calculation and rounding to 3 significant digits yields the result

C = 605 , 320 ft size 12{C="605","320"` ital "ft"} {}

Let us ponder a second question based upon the data presented in the original problem.

Question 2 : What is the ground distance traveled by the airplane as it moves from its departure point to its cruise altitude?

Solution : Referring to Figure 2, we observe that we must find the length of the adjacent side in order to answer the question. We can use the definition of the tangent to guide our solution.

tan ( 3 0 ) = Opposite side Adjacent side size 12{"tan" \( 3 rSup { size 8{0} } \) = { { ital "Opposite"` ital "side"} over { ital "Adjacent"` ital "side"} } } {}

Denoting the adjacent side by the symbol A , we obtain

A = Opposite side tan ( 3 0 ) size 12{A= { { ital "Opposite"` ital "side"} over {"tan" \( 3 rSup { size 8{0} } \) } } } {}
A = 31 , 680 0 . 0524 size 12{A= { {"31","680"} over {0 "." "0524"} } } {}

After rounding to 3 significant digits, we obtain the solution

A = 604 , 580 ft size 12{A="604","580"` ital "ft"} {}

Inclined plane

Work is an important concept in virtually every field of science and engineering. It takes work to move an object; it takes work to move an electron through an electric field; it takes work to overcome the force of gravity; etc.

Let’s consider the case where we use an inclined plane to assist in the raising of a 300 pound weight. The inclined plane situated such that one end rests on the ground and the other end rests upon a surface 4 feet aove the ground. This situation is depicted in Figure 3.

Object on an inclined plane.

Question 3: Suppose that the length of the inclined plane is 12 feet. What is the angle that the plane makes with the ground?

Clearly, the length of the inclined plane is same as that of the hypotenuse shown in the figure. Thus, we may use the sine function to solve for the angle

sin ( θ ) = 4 12 = 0 . 333 size 12{"sin" \( θ \) = { {4} over {"12"} } =0 "." "333"} {}

In order to solve for the angle, we must make use of the inverse sine function as shown below

sin − 1 ( sin ( θ ) ) = sin − 1 ( 0 . 333 ) size 12{"sin" rSup { size 8{ - 1} } \( "sin" \( θ \) \) ="sin" rSup { size 8{ - 1} } \( 0 "." "333" \) } {}
θ = 19 . 45 0 size 12{θ="19" "." "45" rSup { size 8{0} } } {}

So we conclude that the inclined plane makes a 19.85 0 angle with the ground.

Neglecting any effects of friction, we wish to determine the amount of work that is expended in moving the block a distance ( L ) along the surface of the inclined plane.

Surveying

Let us now turn our attention to an example in the field of surveying. In particular, we will investigate how trigonometry can be used to help forest rangers combat fires. Let us suppose that a fire guard observes a fire due south of her Hilltop Lookout location. A second fire guard is on duty at a Watch Tower that is located 11 miles due east of the Hilltop Lookout location. This second guard spots the same fire and measures the bearing (angle) at 215 0 from North. The figure below illustrates the geometry of the situation.

Depiction of a scenario associated with a forest fire.

Question: How far away is the fire from the Hilltop Lookout location?

We begin by identifying the angle θ in the figure below.

Refined depiction of scenario.

The value of θ can be found via the equation

θ = 90 0 − 35 0 size 12{θ="90" rSup { size 8{0} } - "35" rSup { size 8{0} } } {}
θ = 55 0 size 12{θ="55" rSup { size 8{0} } } {}

So we can simplify the drawing as shown below.

Trigonometric representation of scenario.

Our problem reduces to solving for the value of b .

tan ( 55 0 ) = b 11 miles size 12{"tan" \( "55" rSup { size 8{0} } \) = { {b} over {"11"` ital "miles"} } } {}
b = ( 1 . 43 ) ( 11 miles ) = 15 . 7 miles size 12{b= \( 1 "." "43" \) ` \( "11"` ital "miles" \) ="15" "." 7` ital "miles"} {}

We conclude that the fire is located 15.7 miles south of the Hilltop Lookout location.

Comment: Triangulation is a process that can be applied to solve problems in a number of areas of engineering including surveying, construction management, radar, sonar, lidar, etc.

Exercises

  1. A 50 ft ladder leans against the top of a building which is 30 ft tall. Determine the angle the ladder makes with the horizontal. Also determine the distance from the base of the ladder to the building.
  2. A straight trail leads from the Alpine Hotel at elevation 8,000 feet to a scenic overlook at elevation 11,100 feet. The length of the trail is 14,100 feet. What is the inclination angle β in degrees? What is the value of β in radians?
  3. A ray of light moves from a media whose index of refraction is 1.200 to another whose index of refraction is 1.450. The angle of incidence of the ray as it intersects the interface of the two media is 15 0 . Sketch the geometry of the situation and determine the value of the angle of refraction.
  4. One-link planar robots can be used to place pick up and place parts on work table. A one-link planar robot consists of an arm that is attached to a work table at one end. The other end is left free to rotate about the work space. If l = 5 cm, sketch the position of the robot and determine the ( x, y ) coordinates of point p ( x , y ) for the following values for θ: (50˚, 2π/3 rad, -20˚, and -5 π/4 rad).

One-link planar robot.

Questions & Answers

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When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
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What is different between quantity demand and demand?
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Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
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Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
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In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
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Answer
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the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
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suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
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types of unemployment
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What is the difference between perfect competition and monopolistic competition?
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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