Define two variable: alpha = 5 pi/8, and beta = pi/8. using these variable, show
ID: 3781310 • Letter: D
Question
Define two variable: alpha = 5 pi/8, and beta = pi/8. using these variable, show that the following trig identity is correct by calculating the values of the left and the right sides of the equation sin alpha cos beta = 1/2 [sin(alpha - beta)+sin(alpha + beta)] in the triangle shown a = 9 cm, b = 18 cm, and c = 25 cm. Define a, b, and c as variables, and then: Calculate the angle alpha (in degrees) by substituting the variables in the Law of Cosines. (Law of Cosines: c^2 = a^2 +b^2-2abcosgamma) Calculate the angles beta and gamma (in degrees) using the Law of Sines. Check that the sum of the angles is 180 degree. In the right triangle shown a = 16 cm and c = 50 cm. Define a and c as variables, and then: Using the Pythagorean Theorem, calculate b by typing one line in the MATLAB Command Window. Using b from part (a) and the a cos d function, calculate the angle a in degrees by typing one line in the MATLAB Command Window. The distance d from a point (x_0, y_0 z_0) to a plane Ax + By +Cz +D = 0 is given by d = |ax_0 + By_0 +Cz_0 + D|/squareroot A^2 +B^2 + C^2 Determine the distance of the point(8,3,-10) from the plane 2x +23y+13z-24 = 0.First define the variables A,B,C,D,x_0,y_0, and z_0 and then calculate d.(Use the abs and sqrt functions.)Explanation / Answer
PROBLEM 2:
I have taken pi value as 180.
>> a = (5*180)/8
a =
112.5000
>> b = (180/8)
b =
22.5000
>> LHS = sin(a) * cos(b)
LHS =
0.4912
>> RHS = 0.5 * (sin(a-b) + sin(a+b))
RHS =
0.4912
We can see LHS = RHS.
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PROBLEM 3:
(a)
To find alpha, the Law of Cosines formula used is as below.
I have denoted alpha as 'x'.
a2 = b2 + c2 -2bccosx
cosx = (b2 + c2 - a2)/(2*b*c)
To get x, take cos inverse of RHS
>> a
a =
9
>> b
b =
18
>> c
c =
25
>> X = (b^2 + c^2 - a^2)/(2*b*c)
X =
0.9644
>> X = acosd(X)
X =
15.3341
The alpha value is 15.3341 degrees
(b)
Law of Sines for triangle with sides a, b and c and angle denoted by X (opposite to side a), Y(opposite to side b) and Z(opposite to side c) is
(sin(X)/a) = (sin(Y)/b) = (sin(Z)/c)
Here alpha = X
beta = Y
gamma = Z
To calculate Y and Z using Law of Sines, we use the below formula,
We have a and A available from part (a).
So, sinY = b * (sin(X)/a)
To get Y take sine inverse.
Similarly, sinZ = c * (sin(X)/a)
To get Z take sine inverse.
>> a
a =
9
>> b
b =
18
>> c
c =
25
>> X
X =
15.3341
>> Y = asind(b*(sind(X)/a))
Y =
31.9309
>> C = asind(c*(sind(X)/a))
C =
47.2715
(c)
>> X+Y+Z
ans =
94.5365
Sum of all angles will not be equal to 180 if you find all angles by the formula.
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PROBLEM 4:
(a)
>> a = 16
a =
16
>> b = 50
b =
50
>> c = sqrt(a^2 + b^2)
c =
52.4976
(b)
alpha = acosd(b/c)
alpha =
17.7447
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