Use Cosine

8/11/2019 Use Cosine

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Vector: a quantity that has a magnitude as well as direction

Simple Trig: Law of Sines, Law of Cosines, and (SohCahToa) rules

To Begin: Watch this video as a thorough explanation of the topic! If it helps, it is recommended that you follow along with the running example throughout

the video. This way you have hard notes and a decent example to look back at.

CLICK ME!!!36) Calculate the sum of vectors using law of cosines and component methods.

http://screencast-o-matic.com/watch/clf1elkuW





Let's sum up the Component Method in words and Simple Steps!



http://phet.colorado.edu/en/simulation/vector-addition - use this tool that was available to us to practiceand understand the component method better

Now Let's Sum up Law of Cosine in a few simple steps!

http://phet.colorado.edu/en/simulation/vector-addition







NOW LET'S PRACTICE SOME PROBLEMS!

Let's Try an Example of Both Law of Cosines and Component Method.This written out may help you understand.

Law of COSINE

ANSWER

ANGLE TIMEChallenge Alert! Try Component Method on your own. The answers are given above using Law of

Cosine, so you are able to check. Look at the steps above, or the video explanation if you need

help.



CHOOSE WHICH METHOD IS BEST FOR OUR NEXT EXAMPLE.

Please try befor e reading on.

YOU GUESSED IT!!

Component Method would be the optimum way to solve for the Resultant Vector.

TIP: Many people get hung up on finding the right triangles while doing component method. If you

can't visualize it, then just focus on the components. This is what helped me the most in thissection. Look above at component steps for help with the equations.

This problem is a good example of simply using the components without a confusing picture. In this case,

there are three sides (components) added together.

Angle Time:



Portfolio Problems

These problems aren't meant to be extremely challenging, the point is to gain understanding.

One problem using Law of Cosine

One problem using Component Method



X- and Y

-Components of a Force Vector

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This artic le discusses the x- and y-

components o f a force vector . Understand

that the diagrams and mathematics here could be applied

to any type of vector such as a displacement, velocity, oracceleration vector.

For an understanding of vectors see the Vectors sectionof the Physics Department.

For an understanding of r ight tr iang le tr igonometry seethe Trigonometry and Right Triangles section of the

Trigonometry Realms.

When you are finished with the material here, be sure tovisit the Force Component Machine. It will show you howthe positive and negative signs for the force componentswork for any direction that the two dimensional forcevector may be pointing.

The two dimensional force vector

A force vector can be expressed in two d imens ions onthe (x, y) plane . For example, imagine the surface of a

table top to be an (x, y) plane. Objects can be pushedacross this table surface in several different directions, not just parallel to the length or width of the table. Objects canbe pushed across a table top at a slanted direction relativeto the edges of the table top. In the animation below wesee several different directions in which you could push an





http://zonalandeducation.com/mmts/trigonometryRealms/introduction/rightTriangle/trigRightTriangle.html




http://zonalandeducation.com/mstm/physics/mechanics/forces/forceComponentMachine/forceComponentMachine.html










object across a table top, or the several directions one canapply a force to an object on an (x, y) plane. The objectbeing pushed is the green disk, and the force vector is the

black arrow:

Force vectors like the one shown above are said to be twodimensional force vectors. You can think of them as forcesthat have a part that pushes r ight o r lef t , and that haveanother part that pushes up or down . These parts of the

force are called the components of the force. Thecomponent that pushes right or left is called the x-

component , and the part that pushes up or down is calledthe y-component .

Force com ponents and shadows

Mathematical ly , the components act like shadows of theforce vector on the coordinate axes.



In the picture directly below we see a force vector on the(x, y) plane. The force vector is white, the x-axis is red, they-axis is green, the origin is white. It is common to position

force vectors like this with their tails at the origin. The lightin this picture is shining directly into the (x, y) plane, andwe see no shadows from this view. For our purposes herethe axes and vector are drawn unusually wide; they arenormally drawn as thin lines in diagrams.

The vector on the (x, y) plane

Right below is the same scene from another viewpoint.The light is now shining directly from above. That is, thelight is shining straight down parallel to the y-axis. Note



the shadow of the vector on the x-axis. This shadowrepresents the x-component of the force vector.

The x-component

Next, below, we have the same situation except thedirection of the light has changed. The light now is shiningfrom the right, parallel to the x-axis. A shadow of the forcevector can be seen on the y-axis. This shadow,mathematically, is the y-component of the force vector.



The y-component

Force vector component diagrams

We are back to a flat surface diagram below; it shows howthese components can be drawn.

The black vector is the two dimensional force vector ,labeled F.

The red vector is the x-component of the force vector,labeled Fx. It would be pronounced 'F sub x'. Since 'x' isactually a subscript, this notation usually looks like this:



However, in Zona Land Education the subscript's position

is often implied, as here, hopefully without any loss ofmeaning.

The green v ector is the y-component of the force vector,labeled Fy, pronounced 'F sub y'.

The components of the force vector can also be arrangedthis way, forming a r ight tr iang le :



Force vector component mathemat ics

If we know the size of the two dimensional force vector,the black one in the above diagram, and the angle itmakes with the x-axis, then we can use right triangletrigonometry to find the values for the components.

In the following diagram 'A' is the angle that the twodimensional force vector makes with the x-axis. Usingright triangle trigonometry, Fx is adjacent to angle A, Fy is

opposite to angle A, and F is the hypotenuse, as:









Unusual diagram

The above diagram shows how the trigonometry is usuallypresented - the cos ine funct ion is associated with the x-

component and the s ine funct ion is associated withthe y-component . However, it is not the only way to thinkabout it. The following is a legitimate vector diagram forthis force vector, but the x-component is calculated withthe sine function, the y-component with the cosine. Notewe are using angle B now; it's a different angle:



Remember, the diagram and formula derivation above,although correct within its own context, is unusual as farcommon textbook examples are concerned. They areusually set up to solve for the x-component using thecosine function and the y-component using the sine, aswas presented originally with angle 'A'. There is goodreason for this. If the direction of the force vector is given

in standard position, as angle A could be interpreted, thenthe original derivations give correct results.



The sign o f the components

The x-component of the force vector can be positive or

negative. If it points to the r ight , it is posi t ive . If it points to the left , it is negative .

The y-component of the force vector can be positive ornegative.

If it points up , it is posi t ive . If it points down , it is negat ive .

When right triangle trigonometry is used, you need toconsult your vector diagram to decide which way thecomponents are pointing and then assign the correct signto your calculated values as a last step in your solution.The right triangle trigonometry as presented here will

always yield posi t ive results. It is really only solving forthe lengths of the legs of a right triangle, as one might doin Geometry studies.

Is F ever negative?

F, the value for the size of the two dimensional forcevector, is always stated as a positive number.



Prob lem examp le

An object is pushed across a table top with a force of 16.6

N directed 32.7 degrees S of E. What are the x- and y-components of this force?

Here is the diagram:

This is the solution:

Solving for the x-component , Fx :

Right triangle trigonometry: The cosine of an

acute interior angle of a right triangle is equal tothe length of the adjacent side to the angle

divided by the length of the hypotenuse.



Use the length of the force vectors as the lengthsof the adjacent side and hypotenuse. Fx is the

adjacent side, F is the hypotenuse.

Algebra: Multiply both sides by F.

Algebra: F's cancel on the right side.

Algebra: Switch left and right sides.

Plug in values.

Cosine of 32.7 degrees is 0.8415.

Multiply, result stated in significant figures.

Final result stated with units. Note the positive

value of the final result. It is positive because thisis an x-component, and in the diagram it is aimed

to the right. As usual with numbers, the plus sign

is optional.

Solving for the y-component , Fy :

Right triangle trigonometry: The sine of an acute

interior angle of a right triangle is equal to thelength of the opposite side to the angle divided by

the length of the hypotenuse.



Use the length of the force vectors as the lengths

of the opposite side and hypotenuse. Fy is theopposite side, F is the hypotenuse.

Algebra: Multiply both sides by F.

Algebra: F's cancel on the right side.

Algebra: Switch left and right sides.

Plug in values.

Sine of 32.7 degrees is 0.5402.

Multiply, result stated in significant figures.

Final result stated with units. Note the negative

value of the final result. It is negative becausethis is an y-component, and in the diagram it is

aimed down.

Always remember to check back to the diagram to makesure you have the sign of the components correct.

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