c = 3.6 yd, 7.8 yd

UpStudy Free Solution:

To find the third side \(c\) of the triangle with given \(A = 29.7^ \circ \), \(a = 3.9 \space \text { yd} \), and \(b = 6.4 \space \text { yd} \), we can use the Law of Cosines. The Law of Cosines states:

\(c^ 2 = a^ 2 + b^ 2 - 2ab \cos ( A) \)

Let's plug in the given values and solve for \(c\):

1. \(a = 3.9 \space \text { yd} \)

2. \(b = 6.4 \space \text { yd} \)

3. \(A = 29.7^ \circ \)

First, we convert the angle \(A\) to radians, but since most calculators can handle degrees directly for cosine, we can proceed directly:

\(c^ 2 = ( 3.9) ^ 2 + ( 6.4) ^ 2 - 2( 3.9) ( 6.4) \cos ( 29.7^ \circ ) \)

Calculating each term:

\(( 3.9) ^ 2 = 15.21\)

\(( 6.4) ^ 2 = 40.96\)

\(2 \times 3.9 \times 6.4 \approx 49.92\)

Next, calculate the cosine of \(29.7^ \circ \):

\(\cos ( 29.7^ \circ ) \approx 0.86863\)

Now plug these values into the equation:

\(c^ 2 = 15.21 + 40.96 - 49.92 \times 0.8686\)

\(c^ 2 = 15.21 + 40.96 - 43.36 \)

\(c^ 2 = 12.81\)

\(c = \sqrt { 12.81} \approx 3.6 \space \text { yd} \)

Additional Possible Solution using Law of Sines

We need to check if there is another solution due to the ambiguity in the Law of Sines for non-right triangles, which happens when there are two possible values for an angle:

Using the Law of Sines:

\(\frac { a} { \sin ( A) } = \frac { b} { \sin ( B) } = \frac { c} { \sin ( C) } \)

First, find angle \(B\):

\(\sin ( B) = \frac { b \sin ( A) } { a} = \frac { 6.4 \sin ( 29.7^ \circ ) } { 3.9} \)

\(\sin ( B) = \frac { 6.4 \times 0.4954} { 3.9} \approx 0.8129\)

To find the angle \(B\) from the value \(\sin ( B) = 0.8129\), we will use the inverse sine (arcsine) function.

\(B = \sin ^ { - 1} ( 0.8129) \)

Using a calculator to find the angle in degrees:

\(B \approx \sin ^ { - 1} ( 0.8129) \approx 54.38^ \circ \)

So, the angle \(B\) is approximately \(54.38^ \circ \).

Let's recompute \(C\) and verify our previous steps:

1. Given:

\(A = 29.7^ \circ , \quad B \approx 54.38^ \circ \)

2. Calculate angle \(C\):

\(C = 180^ \circ - A - B = 180^ \circ - 29.7^ \circ - 54.38^ \circ \approx 95.92^ \circ \)

Using the Law of Sines to find \(c\):

\(\frac { a} { \sin ( A) } = \frac { c} { \sin ( C) } \)

\(c = \frac { a \sin ( C) } { \sin ( A) } \)

\(c = \frac { 3.9 \sin ( 95.92^ \circ ) } { \sin ( 29.7^ \circ ) } \)

Calculate the sine values:

\(\sin ( 95.92^ \circ ) \approx 0.9946\)

\(\sin ( 29.7^ \circ ) \approx 0.4954\)

Now, calculate \(c\):

\(c = \frac { 3.9 \times 0.9962} { 0.4961} \approx \frac { 3.8842} { 0.4961} \approx 7.8 \space \text { yd} \)

Supplemental Knowledge

The Law of Cosines is a powerful tool in trigonometry used to find the lengths of sides or the measures of angles in any triangle, especially when dealing with non-right triangles. The formula is:

\(c^ 2 = a^ 2 + b^ 2 - 2ab \cos ( C) \)

where:

- \(a\), \(b\), and \(c\) are the sides of the triangle.

- \(C\) is the angle opposite side \(c\).