0.13 Phy1100: force -- vector solutions for coplanar forces concurrent  (Page 4/13)

Let's give that scenario a test drive

Pretend that all we know about the triangle is the following:

• Side a = 10
• Side b = 11.3741339
• Angle B = 80 degrees
• sin B = 0.984807753

We need to solve for angles A and C as well as side c.

Substitution of known values

Recall that the general formula for the law of sines is:

(a/sin A) = (b/sin B) = (c/sin C)

Substitution of the known values into the formula gives:

(10/sin A) = (11.3741339/0.984807753) = (c/sin C)

Rearranging terms

Using the first two ratios and rearranging terms gives us:

sin A = 10*0.984807753/11.3741339 = 0.865830983

A = arcsine(0.865830983) = 1.04680884 radians = 60 degrees

(Note that I used an extreme number of decimal digits to avoid rounding errors and cause the final answer to come out almost exactly correct.)

Compute the value of angle c

Now that we know the values of angles A and B, we also know that

Angle C = 180 - 80 - 60 = 40 degrees.

Compute the length of side c

Using the second and third ratios and rearranging terms, we can solve for side c as:

c = sin C * (11.3741339/0.984807753)

= 0.64278761 * (11.3741339/0.984807753)

= 7.42393866

Thus, we have determined that:

• Angle A = 60 degrees
• Angle C = 40 degrees
• Side c = 7.42

These values all match what we know to be true from the first example.

The law of cosines

In addition to the law of sines, there is also a law of cosines. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known. The law is also useful in computing the angles of a triangle when all three sides are known.

Using the same names for the sides and the angles, there are three formulas for the law of cosines :

• c^2 = a^2 + b^2 - 2*a*b*cos C
• a^2 = b^2 + c^2 - 2*b*c*cos A
• b^2 = a^2 + c^2 - 2*a*c*cos B

Compute the length of side c

Using the known values from the earlier example, we can compute the length of side c by entering the following expression into the Google search box:

sqrt(10^2 + 11.3^2 - 2*10*11.3*cos(40*pi/180))

This produces an answer of 7.39, which is close enough to the known value for the length of side c.

Compute angle C in degrees

Similarly, using the known values from the earlier example, we can compute the angle C knowing the lengths of the three sides. Enter the followingexpression into the Google search box:

arccos((10^2 + 11.3^2 - 7.39^2)/(2*10*11.3))*180/pi

This produces an answer of 40.02 which is close enough to the known value for the angle C in degrees.

An exercise for the student

I will leave it as an exercise for the student to work through the algebra to determine how I used the known values and the first formula given above to construct the expressions that I entered into the Google search box.

Discussion and sample code

Now that we have expanded our knowledge of trigonometry, let's examine some solutions to problems involving force using vectors.

All of the exercises that we will work through in this module involve forces in a two-dimensional plane (coplanar) that meet at a point (concurrent at apoint). This excludes forces that create moments or torques, which will be the topic of future modules, and also excludes forces that may come from anydirection in a third dimension.