Trigonometric and vectors PDF

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An experiment about trigonometric and vectors ...

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Experiment II

Physics I, #Mondays, #Physics Laboratory report Laboratory experiment II, Trigonometry, and vectors I. Introduction The main objective of Physicists uses mathematics as a language that benefits them to express and act with physical systems. Because we live in a three-dimensional universe, we demand to use trigonometry as an instrument to picture it. We also need vectors if we want to explain how we move around such a universe. The following experiment will be studied based on Trigonometric functions, properties of triangles, and vector manipulations. References Robert Ellis and Denny Gulick, Calculus with Analytic Geometry, Second Edition, Harcourt Brace Jovanovich Inc., NY, 1983. Theory Trigonometric functions In this trigonometric function problem, a right triangle is a geometrical form that has three sides (a, b, c) that intersect at three vertexes (A, B, C), creating three angles (α, β, γ), one of which is 90°, as shown in Figure 2.1. In this situation, the side opposite to the 90° angle has named the hypotenuse.

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Experiment II

Figure. 2.1 Right triangle Presented a right triangle, we determine the trigonometric functions sine, cosine, and tangent of an angle as follows,

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒

Sine=

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

,

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒

Cosine=

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

,

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒

Tangent=

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒

Applying these relationships to the triangle in Figure 2.1,

𝑎 𝑐

sin𝛼=

cos𝛼 =

𝑏 𝑐

𝑠𝑖𝑛𝛼 𝑎 𝑏

tan𝛼 = 𝑐𝑜𝑠𝛼=

sin𝛽 =

𝑏 𝑐

𝑎

cos𝛽 = 𝑐 tan=

𝑠𝑖𝑛𝛽 𝑏

𝑐𝑜𝑠𝛽 𝑎 =

2. Properties of triangles For any triangle, the following relationships can be mathematically proven (using the notation from Figure 2.2), c2=a2+b2-abcos𝛾

𝑏 𝑐 𝑎 = = 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛽 𝑠𝑖𝑛𝛾

(2.1) ( 2.2)

A

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Experiment II

C

B

a Figure. 2.2 Triangle Please see that the right triangle in Figure 2.1 is a particular situation of the triangle shown in Figure 2.2 with γ = 90°. For the right triangle, equations 2.1 and 2.2 become, c2=a2+b2-2abcos90°=a2+b2 which is understand as the Pythagorean Theorem, and,

𝑏 𝑐 𝑏 𝑎 𝑎 = = = c ⇒ sin𝛽= ; sin𝛼 = 𝑐 𝑐 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛽 𝑠𝑖𝑛 90°

which agree with the definitions of sine and cosine given in the previous section. 3. System of coordinates A method of coordinates in n-dimensions consists of n axes perpendicular to each other. A 2dimensional way, then, consists of 2 axes, usually called x and y, that cross each other at a point called O or origin. It is expected to label each axis as shown in Figure 2.3, with the positive xaxis looking to the right and the positive y-axis facing upward.

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Experiment II

A point P in a 2-dimensional space, or plane, is defined by an ordered set of 2 numbers (xp, y2) in Cartesian coordinates or (Rp, θp) in Polar coordinates, as shown in Figure 2.4. The Cartesian coordinates xp and yp are equal to the distance from the origin O to point P measured along the axes.

Note that each point is represented by a single pair of ordered numbers. Conversely, each assigned pair of numbers recognizes a unique point. To translate the coordinates of the point from one system to the other, we remark from Figure 2.4 that the portion that goes from O to xp, the vertical line that goes from P to the horizontal axis and RP form a right triangle. Utilizing the parts of a triangle we receive,

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Experiment II

Given (Rp, θp),

𝑦𝑝 ⇒ yp= Rpsin𝜃p 𝑅𝑝

sin𝜃p=

𝑦𝑝 ⇒ yp= Rpcos𝜃p 𝑅𝑝

cos𝜃p= Given (xp, yp):

Rp= √𝑥𝑝2 + 𝑦

2 𝑝

𝑦𝑝 𝑥𝑝 )

𝜃𝑝 = tan-1(

4. Scalars and vectors Conceptually, a scalar is a quantity that just has a module or magnitude. A vector is a number that has a module or magnitude and a direction. Graphically, vectors are represented by arrows. The length of the arrow gives the magnitude of the vector, and the direction is the angle between the arrow and the positive horizontal axis.

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Experiment II

Figure 2.5 presents a vector, where we have utilized the conventional notation: the vector is indicated by a bold letter A, its magnitude by A, and its direction by θA. Angles are always measured in the counterclockwise direction from the positive horizontal axis. 5. Adding vectors a. Graphical i. (Very important) Head to tail method Also, two vectors, move one of the vectors parallel to itself until its tail is at the tip of the other vector. The vector total C = A + B is the vector that goes from the tail of the first vector to the top of the second one. See Figure 2.6 (a).To add more than 2 vectors, they require be placed ”head to tail” in order. The vector total proceeds from the tail of the first vector to the tip of the last one. See Figure 2.6 (b). The order does not change the result.

ii. Parallelogram method

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Experiment II

In order to add two vectors, a parallelogram is created with each vector as a side. The diagonal is the vector sum. See Figure 2.7.

b. Analytical In Figure 2.8, vectors A and B have been added using the “Head to Tail” system. A method of coordinates with the origin coincident with the tail of vector A is also shown in the same Figure.

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Experiment II

Any vector can be signed in terms of its components using trigonometric functions and triangle parts as follows,

𝜃𝐴 By = B sin 𝜃𝐴

Ax = A cos 𝜃𝐴 Bx = B cos However, from the figure we can see that,

𝜃𝐵

Ay = A sin

Cx = Ax + Bx Cy = Ay + By Therefore, we can discover the magnitude and direction of the vector sum C as, C= √𝐶𝑥 + 𝐶𝑦 2

2

𝐶𝑦 ) 𝐶𝑥

𝜃𝐶 = tan-1(

Remark that we initially add component to component. Then we obtain the module or magnitude and direction of the vector sum. This technique can be increased to more than two vectors, as shown in Figure 2.9.

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Experiment II

Dx = Ax + Bx + Cx Dy = Ay + By + Cy D = √𝐷𝑥 + 𝐷𝑦 2

2

𝐷𝑦 ) 𝐷𝑥

𝜃𝐷 = tan-1(

5. Scale

While calculating vectors graphically, they are described by arrows. The length of a vector arrow must be drawn proportional to the magnitude of the vector. That is, vectors must be drawn to scale. The length scale is arbitrary and should be decided such that the vector graph

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Experiment II

satisfies the graph paper. The scaling factor must be clearly indicated on the graph in such a way that anyone can read it. The graph must be done correctly because the length and angle of the vector sum must be read straight from it. V. Experimental procedure 1. Trigonometry and properties of triangles. Sketch to scale right triangles with one of the angles equal to 30°; 45° and 60°. Estimate the length of their sides. Later, using the triangles and the determinations of trigonometric functions, determine the sine, cosine, and tangent of 0°; 30°; 45°; 60° and 90°. Show your results in a table. Determine the percent error.

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Experiment II

0°

30°

45°

sin

0

1/2

cos

1

√3 2

tan

0

1

60° √3 2

1

√3

√2 1

√2 1

90°

1/2

0

1 √3

Undefined /∞

2. Vector addition.

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Experiment II

a. Address a vector with magnitude A = 5.0 m and direction θA = 60°, determine its components Ax and Ay analytically and graphically. Find the percent error.

𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟

Graphical 𝐴𝑥 = ___2.5___

Analytical 𝐴𝑥 = ___2.5____

%error 𝐴𝑥 = _____0%_____

=

𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

𝐴𝑦 = ___4.3_

×100%

|𝐸−𝐴|

=

𝐴

×100%

𝐴𝑦 = ___4.33__

%error 𝐴𝑦 = _____0.69%_____

|2.5−2.5|

𝐴𝑥 =

2.5

×100

= 0%

%error 𝑨𝒙 = __0%__ 12

Experiment II

|4.3−4.33| ×100 = 0.69% 4.33

𝐴𝑦 =

%error 𝑨𝒚 = _0.69%_

b. Provided two vectors with magnitudes A = 2.0 m and B = 2.0 m and directions 𝜃𝐴 = 30° and 𝜃𝐵 = 120°, find their vector sum C = A + B analytically and graphically. Find the percent error.

Graphical: C = _2.82_____ 𝜃𝑐 = __44°_____

Analytical: C = ___2.8___ 𝜃𝑐 = __45°_____

|2.8−2.82|

𝐶=

2.82

×100

= 0.7%

%error C = __0.7%____

|44°−45°_|

𝐶=

45°_

×100

= 2.2%

%error 𝜃𝑐 =___2.2%___

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Experiment II

c. Provided three vectors with magnitudes A = 4.0 m; B = 5.0 m; and C = 6.0 m, and directions 𝜃𝐴 = 30°; 𝜃𝐵 = 90°; and 𝜃𝑐 = 225°, find the vector sum D = A + B + C graphically and analytically. Find the percent error.

Graphical: D = __2.9____ 𝜃𝐷 = __100°_____

Analytical: D = ____2.87__ 𝜃𝐷 = __106°____

|2.9−2.87|

𝐷=

2.87

×100

= 1.0%

%error D = ___1.0%___

|100°−106°|

𝐷=

106°

×100

= 5.7%

%error 𝜃𝐷 =___5.7%___

d. Name the inverse of a vector as another vector of the equivalent magnitude but looking in the opposite direction. Provide two vectors with magnitudes A = 2.0 m and B = 2.0 m, and directions 14

Experiment II

A= 45° and B= 135°, find the magnitude and direction of C such that A + B + C = 0. Calculate by analytically and graphically. Find the percent error.

Graphical

C=___5.7___ 𝜃𝑐 =_270°_____

Analytical

C=__5.64____

𝜃𝑐 =__270°____

|𝐸−𝐴| ×100% 𝐴

Percent error =

C=

%error

%error

|5.7−5.64| ×100 = 1.1% 5.64

C =____1.1_____

𝜃𝑐 =_____0____

𝜃𝐶 =

|270−270| 270

×100

= 0%

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Experiment II

Conclusion In conclusion, by measuring and calculating all the data graphically and analytically we can assume that when we are working an experiment graphically or analytical, must of the time lose a percentage of a specific experiment. Considering the percent error to assure if the estimation we are doing is exact. Also, considering how far it's getting accurate data. In addition, after all the calculations are collected it simply identifies how physicists use math when connecting to their work, and how vectors are used to describe the directions.

Questions 1. Prove that sin2α + cos2 α = 1 for all α. (Hint: use triangle properties). Using the Pythagorean theorem we know that, a2+b2 = c2 sin2α + cos2 α = 1 Sinα =

Cosα =

𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑏𝑎𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

Perpendicular2 + base2= hypotenuse2

𝑏𝑎𝑠𝑒 2 2 ) =1 (𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ) +( ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

sin2α + cos2 α =

2. Can a vector have a component greater than its magnitude? Explain. No. It cannot.

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Experiment II

A component is a projection of a vector in a particular direction. It is like a part of a whole(vector). At most it can have the same magnitude. In that case the vector has to be in the direction along which the component was taken.

3. If A + B = 0, what can you say about the components of the two vectors? Equal and opposite/ One positive and one negative. 4. Can a vector have a component equal to zero and still have a non-zero magnitude? Explain. Yes, if one is zero and one is nonzero 5. Does the order in which vectors are added make a difference when graphical or analytical methods are used? Explain The order in which the vectors are added make a difference when graphical or analytical methods are used, this is true. Yes, because the word vector itself indicate directions

6. Using the definition of “opposite vector”, show that the subtraction of vector A from vector B equals the sum of vector B plus the opposite of vector A. The opposite tor of A is -A and thus vector B-A= B+(-A)

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