Question

7a)
in △ abc,
a = 38°, b = 27, c = 34
find a

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides and the included angle (SAS), so we use the Law of Cosines. The formula for the Law of Cosines when finding side \(a\) is \(a^{2}=b^{2}+c^{2}-2bc\cos A\).

Step2: Substitute the Given Values

Given \(A = 38^{\circ}\), \(b = 27\), \(c = 34\). Substitute these into the formula:
\(a^{2}=27^{2}+34^{2}-2\times27\times34\times\cos(38^{\circ})\)
First, calculate \(27^{2}=729\), \(34^{2}=1156\), and \(2\times27\times34 = 1836\).
Then, \(\cos(38^{\circ})\approx0.7880\).
So, \(a^{2}=729 + 1156-1836\times0.7880\)
\(a^{2}=1885-1836\times0.7880\)
Calculate \(1836\times0.7880\approx1446.77\)
Then, \(a^{2}=1885 - 1446.77=438.23\)

Step3: Find the Value of \(a\)

Take the square root of \(a^{2}\) to find \(a\): \(a=\sqrt{438.23}\approx20.93\)

Answer:

\(\approx20.93\)