Bearings And Azimuths 5h45c

BEARINGS AND AZIMUTHS BEARINGS AND AZIMUTHS  Horizontal angles are angles measured in a horizontal plane by a level instrument  Used to determine: the locations of points the orientation of lines  The units of measurement most commonly used in surveying is the sexagesimal system based on degrees, minutes, seconds

TYPES OF ANGLES  Interior angles located on the inside of a closed polygon may be measured clockwise or counterclockwise, but the direction must be consistent throughout  Deflection angles are measured right or left from an extension of the current line to the next station are indicated by a numerical value and a "L" or "R"

DIRECTIONS OF LINES  Defined by a horizontal angle between the line and a defined reference line called a meridian  True meridian is the north-south reference line through the earthÆs geographic poles  Magnetic meridian is a north-south reference line as defined by the earthÆs magnetic field  An assumed meridian can arbitrarily chosen to serve as a reference for all other lines in a survey

BEARINGS  Designate the direction of a line by an angle and quadrant letters. (e.g. N30° E)  Bearings are never greater than 90°  Bearings are referenced from north or south and the angle to the east or west from the northsouth meridian.  True bearings are based on true north.  Magnetic bearings are based on magnetic north.

TRUE BEARINGS

BEARING DIRECTION

MAGNETIC BEARINGS

AZIMUTHS    

Angles measured clockwise from any reference meridian Azimuths range from 0 to 360° Azimuths are referenced from north True azimuths are based on true north

 Magnetic azimuths are based on magnetic north

Web Exclusive: Calculating the Direction of a Line using Azimuths November 21, 2000 Wesley G. Crawford RPLS Reprints 11 2 Comments

A breakdown of calculating azimuths. Representing the direction of a line with bearings is a long-standing tradition that continues

today on plats, legal descriptions, site plans, and other documents. Calculating the direction of a line by using bearings is also a longstanding tradition. However, some surveyors have realized that calculating the direction of a line is much easier when using azimuths and then converting the azimuths to bearings for representation on documents. I find while presenting training seminars that many people do not know how to calculate directions with azimuths as they have never seen it done. In this article, I will present what azimuths are and how they are calculated.

FIGURE 1 Comparison between Azimuths and Bearings

Azimuths are measured clockwise from North and vary from 0° to 360°. They require only a numerical value (35°, 145°, 215°, etc.). Azimuths may be referenced to true North, to magnetic North, to an azimuth taken from plans, or from an assumed direction. In addition, azimuths may be forward or backward.

TABLE 1

Bearings are measured either clockwise or counterclockwise from North and South. They vary from 0° to 90° and require two letters and a numerical value (N 60° E, S 30° W, etc.). Bearings may be referenced to true North, magnetic North, to a direction taken from plans, or from an assumed direction. Bearings may also be forward or backward.

After traverse angles have been adjusted to represent a closed geometric figure and a starting direction has to be determined or assumed, the direction of each line can be calculated. As noted, this is accomplished using either azimuths or bearings. It is purely a matter of personal preference because they both represent the same thing—a direction. However, as stated earlier, azimuths are much easier to calculate. Since they are easier, fewer mistakes in calculations generally occur. It will also be shown how bearings can easily be obtained by converting from azimuths.

Determining the Quadrant

By observation of the value of the azimuth, the surveyor should be able to instantly visualize where the azimuth is located. Whether it is in the Northeast quadrant, Southeast, Southwest, or Northwest. Obtaining a Back Azimuth

To calculate azimuths about a traverse, it is necessary to obtain the back azimuth of a line. To calculate a back azimuth, simply add 180° to the azimuth of the line. Back Azimuth = Forward Azimuth + 180° For example, if a line has an azimuth of 75, its back azimuth would be 255°. If a line has an azimuth of 150, its back azimuth would be 330.

Calculating Directions Clockwise or Counter-Clockwise

Direction calculations can be performed in either a clockwise or counter-clockwise manner about a traverse. There are specific rules for calculating in either direction. To calculate azimuths clockwise around a traverse: Subtract the interior angle from the back azimuth of the preceding line.

To calculate azimuths counter-clockwise around a traverse: Add the interior angle to the back azimuth of the preceding line.

Step-by-step Procedure for Calculating Directions

The following is the step-by-step procedure for calculating azimuth of line BC for the figure shown in Steps 1 through 4. Step 1

Plan and prepare. Determine a known azimuth. In this case, it is shown to be 45 and the direction that the calculation will proceed (clockwise or counterclockwise). In this case, we are going to calculate clockwise from A to B to C. List the adjusted interior angle draw a sketch of the entire traverse. Be sure your sketch is reasonably accurate (that is, angles and distances should be close to scale). Label the points, the starting direction, and the interior angles. Orient the drawing properly to North.

Step 2

Perform the calculation. Start by writing down the starting azimuth. Add 180° to obtain the back azimuth. Subtract the interior angle to obtain the azimuth of the next line. If the result is greater than 360, subtract 360. Write down the azimuth on the sketch. Step 3

Repeat the calculation for each line of the traverse. That is, add 180° and subtract the interior angle.

EQUATION 2 Step 4

Check the calculations by using the last interior angle to recalculate the starting azimuth. Example For the figure below, calculate the direction of each line and provide a check. Proceed clockwise about this traverse. The calculations are shown in a tabular form for ease of understanding along with a sketch of the calculation of each line.

Azimuth of AB = Add 180° to get Back Azimuth ------------------------ Back Azimuth = Subtract Interior Angle ---------------------- The Azimuth of BC is 350°00' +180°00' --------- 530°00' -64°21' ------- 105°39'

Calculation of the Direction of Line BC

Azimuth of BC = Add 180° to get Back Azimuth = --------------------- Back Azimuth = Subtract Interior Angle --------------------- The Azimuth of CD is

105°39' +180°00' --------- 285°39' -206°35' --------- 79°04'

Calculation of the Direction of Line CD

Azimuth of CD = Add 180°00'to get Back Azimuth ------------------ Back Azimuth = Subtract Interior Angle ---------------------- The Azimuth of DE is 79°04' +180°00' -------- 259°04' -64°53' --------194°11'

Calculation of the Direction of Line DE

Azimuth of DE = Add 180°00'to get Back Azimuth ------------------ Back Azimuth = Subtract Interior Angle ---------------------- The Azimuth of EA is 194°11' +180°00' -------- 374°11' -96°38' -------- 277°33'

Calculation of the Direction of Line EA

Azimuth of EA = Add 180°00'to get Back Azimuth ------------------ Back Azimuth = Subtract Interior Angle ---------------------- The Azimuth of AB is 277°33' +180°00' -------- 457°33' -107°33' -------- 350°00'

Calculation of the Direction of Line AB as a Check Summary

That’s all there is to calculating azimuths. Simply follow a simple rule if going clockwise about the traverse: “Subtract the interior angle from the back azimuth of the previous course,” and the calculations can be performed quickly and easily. If bearings are needed, simply convert from azimuths to bearings as shown in the Table at the beginning. For those who aren’t using azimuths, I hope this article helps in understanding why many people now use azimuths. The video directly below will demonstrate how to gain the area of an irregular quadrilateral. For this particular type of quadrilateral, we know the lengths of all four sides and one internal angle. The steps for finding this area can be done by performing these steps: 1. Divide the figure into two triangles by drawing a diagonal. 2. Calculate the area of the triangle that has the given angle. 3. Calculate the length of the diagonal using the Law of Cosines. 4. Calculate the area of the second triangle using Heron's Formula. 5. Add the two areas of the triangles to determine the area of the quadrilateral.

The Law of Sines Home > Lessons > Law of Sines

Search | Updated January 28th, 2017

Introduction These are the sections in this lesson: 

The Law of Sines Equation



Calculate the Length of a Side



Calculate the Measure of an Angle



Choosing the Correct Law



Instructional Videos



Interactive Quizzes



Activities



Related Lessons

Equation: The Law of Sines Triangles are a fundamental geometrical shape. Triangles appear within several disciplines, some of which are architecture, engineering, astronomy, and chemistry. This is why mathematicians have studied them and consequently have several relations to enumerate their sides and angles. Here is one such relation. This is the equation called The Law of Sines.

As it will be thoroughly explained within the next section, this equation is useful when a certain angle and its opposite side are known (and either another side or angle) for a given triangle. If this given information is known, then it is possible to calculate another known angle's opposite side or another known side's opposite angle. If you are not familiar with how to label a triangle or you have never worked with sine, cosine, or tangent, please read our trigonometry basics section before moving on to the

next sections within this page.

Calculating the Length of a Side Let's say we are given the following information for a triangle.

This situation is appropriate for The Law of Sines. The reason it is appropriate is because the 15 degrees and the 7.5 miles are opposite of each other. This is emphasized in the next graphic.

Consequently, we will be able to solve for the side marked with the 'x.' [The angles have been marked with the letters 'A,' 'B,' and 'C.' The labeling is arbitrary and the problem could be solved with any arrangement of the letters.]

The reason side-x can be calculated is because it is opposite a known angle, angle-C.

Now we will refer to our equation. The Law of Sines has three ratios — three angles and three sides.

We have only three pieces of information. We also know nothing about angle-A and nothing about side-a. So, we will only need to utilize part of our equation, which are the ratios associated with 'B' and 'C.'

We know angle-B is 15 and side-b is 7.5. We also know that angle-C is 20, but we do not know side-c. We will substitute these values into the equation as such.

Now that we were able to construct two equal ratios, called a proportion, let's solve it by cross-multiplying.

Cross-multiplying will give us these equal products.

The equal products above make a simple equation. We have to solve for the variable 'c.' To do so, we will have to divide both sides of the equation by the sin 15.

We can cancel the sin 15s because we're both multiplying and dividing 'c' by that value.

With the sin 15 out of the way, we need to plug this into a calculator.

After evaluating with a calculator, this is our final value for 'c.'

This means the value of 'x' in our problem is 'c' also, which is 9.9 miles. Use this video and quiz to reinforce the lesson. ideo: The Law of Sines: Find a Side uiz: Law of Sines: Find a Side

Calculating the Measure of an Angle For our next Law of Sines problem, let's take a look at this given information.

This stuation is ideal for The Law of Sines. We were given a side and it's opposite angle.

Since we also have another side, we can solve for another angle, which has been marked with a 'y' in this graphic.

Angle-y rests opposite the 250, which is a known length of our triangle.

The Law of Cosines Home > Lessons > Law of Cosines

Introduction Here are the sections within this lesson:

Search | Updated January 28th, 2017



The Law of Cosines Equations



Calculate the Length of a Side



Calculate the Measure of an Angle



Choosing the Correct Law



Instructional Videos



Interactive Quizzes



Activities



Related Lessons

Equations: The Law of Cosines Triangles are a fundamental geometrical shape. Triangles appear within several disciplines, some of which are architecture, engineering, astronomy, and chemistry. This is why mathematicians have studied them and consequently have several relations to enumerate their sides and angles. Here is one such relation. These are the equations collectively called The Law of Cosines.

As it will be thoroughly explained within the next section, these equations are useful for two cases: 1) when two sides and the angle between them are known within a triangle and 2) when all three sides of a triangle are known. If either case is known, then the law of cosines is helpful. If you are not familiar with how to label a triangle or you have never worked with sine, cosine, or tangent, please read our trigonometry basics section before moving on to the next sections within this page.

Calculating the Length of a Side

Let's take a look at some given information within the following diagram.

The law of cosines is a helpful tool for this situation because we know a triangle's angle and the sides that are next to it, as emphasized via the colored graphic below.

Now that we know the lawe of cosines can be used for the problem, we now need to label the triangle's angles. The triangle can be arbitrarily labeled with the letters 'A,' 'B,' and 'C.' We will progress with the following arrangement of labeled angles.

Now we can start labeling the sides. The side opposite angle-A, our 35-degree angle, is side-a.

Likewise, side-b is 375, side-c is 400. Using this set of information, we have to determine which law of cosines equation is best to use. Since we know angle-A, we should find the formula that has angle-A in it.

Using this equation, we need to plug in the given information within it.

A calculator can be used to simplify the right side of the equation.

To cancel the square on the right side of the equation, we must employ its inverse function, which is the square root. We will take the square root of both sides of the equation.

Again, with the use of a calculator, we can find side-a by actually using the square root function of a calculator.

It has been shown that the law of cosines is useful for calculating the side of a triangle when we know two sides and the angle between those sides. The side we calculate will always be the side opposite the given angle Use this video and quiz to reinforce the lesson. ideo: Law of Cosines: Find a Side uiz: Law of Cosines: Find a Side

Calculating the Measure of an Angle Here is a given situation involving a triangle.

The given information above is one where we know all three sides of a triangle. Let's label the angles with 'A,' 'B,' and 'C.' Labeling the angles is arbitrary; but, for the sake of continuing along with the given information, we will go with this arrangement of angles.

Recall, we look to the sides opposite sides an angle and label them accordingly. For instance, side-c is opposite angle-C. The graphic below demonstrates this labeling of sides.

It is now clear that side-a is 44, side-b is 29, and side-c is 50. Our next challenge is determine what we can calculate. The only variables that are unknown are angles. So, we must choose which angle to solve for. We can solve for any of them, making our choice an another arbitrary choice. Let's pick angle-C to find.

There are three law of cosine equations to use. Let's review them.

Of the three equations, only one of them contains angle-C. Since we have chosen to solve for angle-C, we will use that equation.

Let's plug in the given information: side-a is 44, side-b is 29, and side-c is 50.

It is now time to start cleaning up the equation so we can solve for angle-C. Let's square the numbers and multiply the factors in front of the cosine function. Doing so will yield this:

The steps that we have taken so far have been basic. It has involved basic the substitution of values, squaring some numbers, and multiplying factors. Now, we have to tackle some algebra. Commonly, students make an error at this step. Some students think these two numbers can be subtracted. However, these numbers cannot be combined because they are not like . Do not make the mistake of combining these values.

Instead of combining unlike , we have to cancel the 2777 by subtracting that value from both sides of the equation.

This is the result of subtracting 2777 from both sides of the equation.

Our next task is to cancel the coefficient. The -2552 must be canceled by dividing that value from both sides of the equation.

Insead of dividing the right side, let's leave the value as a fraction so that we can avoid rounding a decimal value. It's better to round the value once at the conclusion of our calculations. Doing so will allow us to retain the most accurate answer possible.

We have nearly solved our equation for angle-C. However, we have the cosine of angleC. The next step must involve canceling the cosine function. The only way to cancel the cosine function is to use the inverse cosine function, like so.

This is what our equation becomes after we cancel the cosine function with its inverse cosine function.

All that remains now is the use of a calculator. We have to place this information into a calculator to get the final value for angle-C.

The steps above show how we can arrive at an angle when we are given three sides of a triangle. Use this video and quiz to help you reinforce the lesson. ideo: Law of Cosines: Find an Angle uiz: Law of Cosines: Find an Angle

Choosing the Correct Law There are two laws, The Law of Cosines and the Law of Sines (see the Related Lessons section for the lesson). This table will help determine which law to use given the circumstances that are present.

Let it be known to use the Law of Sines we do have to know three pieces of information: 1) we either know two angles and a side or 2) two sides and an angle. When considering the first case, knowing two angles of a triangle means we can find the third angle. If we know a side, we automatically know the angle opposite that side because we know all the angles of the triangle. For the second case, we must know an angle that is opposite one of the two given sides. Having an angle and a side opposite it is the indicator we must use the Law of Sines.

When there are two pairs of opposite sides and angle, we can utilize The Law of Sines. All we have to do now is label the triangle. The triangle below has been arbitrarily labeled with 'A,' 'B,' and 'C.' Any arrangement of 'A,' 'B,' and 'C' can be used; so, we'll stick with this arrangement of letters and angles.

This is when we use The Law of Sines.

We can see that Angle-C is 48 and side-c is 200. Angle-B is unknown, but side-b is 250. So, we will place these values into the equation. Without knowing anything about side-a

and angle-A, we'll use only the ratios that involve 'B' and 'C.'

This is what the equation looks like once the appropriate numbers are plugged inside of it.

When we have two equal ratios, called a proportion, we need to cross-multiply to advance.

This is what the cross-products look like.

To continue solving for B, we have to divide both sides of the equation by 200.

Since we are multiplying and dividing by 200, we can cancel it to get sinB alone.

This is what we are left with, the sin of angle-B.

To cancel the sine of angle-B, we have to utilize the inverse sine function.

Therefore, this is what we need to plug into a calculator.

This is our final answer for angle-B.

Since we were trying to solve for variable-y, it is also 68.3 degrees. Use this video and quiz to reinforce the lesson. uiz: The Law of Sines: Find an Angle uiz: Law of Sines: Find an Angle

Choosing the Correct Law There are two laws, The Law of Sines and the Law of Cosines (see the Related Lessons section for the lesson). This table will help determine which law to use given the circumstances that are present.

Let it be known to use the Law of Sines we do have to know three pieces of information: 1) we either know two angles and a side or 2) two sides and an angle. When considering the first case, knowing two angles of a triangle means we can find the third angle. If we know a side, we automatically know the angle opposite that side because we know all the angles of the triangle. For the second case, we must know an angle that is opposite one of the two given sides. Having an angle and a side opposite it is the indicator we must use the Law of Sines.