EGR 101 LABORATORY 2 APPLICATION OF TRIGONOMETRY IN .

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EGR 101LABORATORY 2APPLICATION OF TRIGONOMETRY IN ENGINEERINGWright State UniversityOBJECTIVE: The objective of this laboratory is to learn basic trigonometric functions,conversion from rectangular form to polar form and vice-versa.EDUCATIONAL OBJECTIVES:After performing this experiment, students should be able to:1.2.3.4.Understand the basic trigonometric functions.Understand the concept of unit circle and four quadrants.Understand the concept of the reference angle.Be able to perform the polar to rectangular and rectangular to polar coordinateconversion.5. Prove few of the basic trigonometric identities.BACKGROUND:Reference Angle: For any given angle, its reference angle is the corresponding acuteversion of that angle. The reference angle is the smallest angle between the terminal sideand the X-axis. Figure 1 shows the concept of a reference angle.TerminalsideθReferenceAngle for θX-axisFigure 1: Concept of Reference angle.Law of cosines:L1L2Өx2 y2

Figure 2: Law of cosines.The squares of the side opposite to the angle is equal to the sum of the squares of the sidesadjacent to the angle minus twice the product of the sides adjacent to the angle and thecosine of the angle.( x 2 y 2 ) 2 l1 l 2 2 * l1 * l 2 * cos θ222 ( x 2 y 2 ) l1 l 2 cos θ 2 * l1 * l 22Law of sines:CbaABcFigure 2: Law of sines.By the law of sines, we have:abc sin( A) sin( B) sin(C )Note: To convert a value in radians to degrees the multiplying factor is 180/pi.PROCEDURE:One-Link RobotI. The unit circle is shown in Figure 3. Consider the vector P of ‘length l’ and ‘angle Ө'. Pis called a Vector as it has both Direction (given by the angle Ө) and a Magnitude (given byl).(X,Y) (0,l)1st QuadrantY2nd QuadrantP (X,Y) (lcosӨ,lsinӨ)lӨX(X,Y) (-l,0)3rd Quadrant4th Quadrant(X,Y) (0,-l)Figure 3: Unit Circle.(X,Y) (l,0)

1. Measure the X and the Y coordinates for the corresponding angles given in theTable 1.2. Find the quadrant in which the point lies.Table 1: Convert from polar to rectangular form and find the reference angle for one link 5270Measuredmagnitudeof XQuadrantMeasuredmagnitudeof YSignof XSignof YReferenceangleThe X and Y values in Table 1 can be written in the form of complex numbers. A complexnumber is any number that can be written as x j*y. This is the rectangular form of thecomplex number.Table 2: Represent the position (X, Y) as a complex number.No.1.2.3.4.5.6.7.The position (X,Y) in Table 1 written as a complex numberFind the values of angle (Ө) and length ‘l’ using the formulasl X 2 Y 2 .(a)YXθ tan 1 ( ) .(b)Table 3: Calculate the length and angle of the one-link ted ‘l’ReferenceAngle

The l and Ө values found in the Table 3 can be represented in the Polar form as P l θ(length, angle from positive x-axis).No.1.2.3.4.5.6.Table 4: Represent the complex number in the polar form.polar form of the position (l and Ө values) found in Table 3II. An identity is a relationship that is true for all permissible values of the variable(s).Verify basic Trigonometric identities given below:1. Take the calculated value of ‘l’ from the Table 3.2. Find the value of sin θ , cos θ , tan θ and sec θ .3. Verify the identities.4. Show the results using a MATLAB program. Example program is given at the endof the lab handout.tan θ sin θcosθ (i)sin 2 θ cos 2 θ 1 (ii)1 tan 2 θ sec 2 θ .(iii)Table 5: To verify the trigonometric identities. (Use the l and Ө values from Table3)CalculatedYXsin θ Cos θ No(X,Y)Tan θSec 0)(-100,0)(-70,-70)Identities(iii)(i)tan θ sin θcosθ(ii) sin 2 θ cos 2 θLHSRHS1 tan 2 θsec 2 θ

Two-links robot:Now consider a two-links planar robot shown in Figure 5. The robot has two joints(shoulder and elbow) and moves in the x-y plane by two motors as shown in Figure 5.I. Similar to the one link robot, find the position of the tip of the robot, P(X, Y) of the twolink robot.WristMotorP (X,Y)L2Θ2L1ElbowMotorΘ1XMotorShoulderFigure 5: Two-link Planar Robot.YP(X, Y)L2Y2Θ2Θ1L1Y1Y1Θ1OX1X2XFigure 6: Tip of the two-link robot represented in the X-Y plane.1. Align the link 2 (L2) to angle Θ2 and then link 1 (L1) to angle Θ1 in accordance withthe angles given in the Table 6.2. Measure, X1, Y1, X2, Y2, X and Y.3. Using the formulas calculate the values of X and Y for the two links robot shown inFigure 6 in MATLAB.

X 1 L1 cos θ1X 2 L2 cos(θ1 θ 2 )Y 1 L1sin θ 1Y 2 L 2 sin(θ 1 θ 2 )X X1 X 2Y Y1 Y 2 X L1 cosθ 1 L 2 cos(θ 1 θ 2 )Y L1sin θ 1 L 2 sin(θ 1 θ 2 )Table 6: Calculate the coordinates of P for a two link planar robot.Angle(Ө1)Angle 4.5.6.7.Measured XMeasured YX1Y1X2Y2MeasuredValuesXYCalculatedValuesXVerify the law of cosines and the law of sine:Y-axisP(X,Y) TipBx2 y2L2180- Ө2OYAαӨ1 βӨ2L1XX-axisFigure 7: Triangle formed by joining the origin O and the tip of the two-link robot.Considering the two-link robot (L1 L2 50mm) shown in Figure 7.Verify the law of cosine and the law of sine for the given values of X and Y in Table 7.Solve for Ө2 and Ө1 using the law of cosines and law of sine.Y

Table 7: To verify the law of cosines and law of sines.No.1.2.Calculated X and YX5575Value ofValue ofӨ2βY7560Lab Report:1. Submit the completed Table 1 through 7.2. Show calculations.Value ofαValue ofӨ1

APPENDIXExample of the MATLAB program for part I:clear allclcdisp('To verify the Trigonometric Identities')%for loop startsfor i 1:1:7% to input the X and Y values.x(i) input('Enter the value of X coordinate: ');y(i) input('Enter the value of Y coordinate: ');%to find the value of rr(i) sqrt(x(i) 2 y(i) 2)% to find sine valuesin(i) (y(i)/r(i))%to find cosine valuecos(i) (x(i)/r(i))%to find tangent valuetan(i) y(i)/x(i)%to find the secant value)sec(i) 1/cos(i)%to display on the screendisp('To verify the identities');%identity 1disp('Identity 1: tan(theta) sin(theta)/cos(theta)');identity 1(i) sin(i)/cos(i)%identity 2disp('Identity 2: sin(theta) 2 cos(theta) 2 1');identity 2(i) sin(i) 2 cos(i) 2%identity 3disp('Identity 3: 1 tan(theta) 2 sec(theta) 2');identity 3 LHS(i) 1 tan(i) 2identity 3 RHS(i) sec(i) 2%for loop endsend

Prove few of the basic trigonometric identities. BACKGROUND: Reference Angle: For any given angle, its reference angle is the corresponding acute version of that angle. The reference angle is the smallest angle between the terminal side and the X-axis. Figure 1 shows the concept of a reference angle.

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