Find the value of 𝑎 to two decimal
places given that the measure of angle 𝐴 is 64 degrees, 𝑐 is 10 centimeters, and
𝑏 is 16 centimeters.
We’ll begin by just adding the
information written in the question to the diagram. We can see that triangle 𝐴𝐵𝐶 is
a non-right-angled triangle in which we’ve been given the length of two of the
sides, the size of one angle and asked to calculate the length of the third
side. As the angle we’ve been given is an
included angle, that means it’s between the two sides whose lengths we know. This is the perfect situation to
use the law of cosines.
The law of cosines allows us to
calculate the length of sides or angles in non-right-angled triangles. And it tells us that if we know the
lengths of two sides and the included angle, then we can calculate the length of the
third side using the formula 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus
two 𝑏𝑐 cos 𝑎. Notice that it must be the angle
opposite the side we wish to calculate which we know, which is the case here.
To calculate the value of 𝑎, we’ll
begin by substituting the length of sides 𝑏 and 𝑐 and the size of angle 𝐴 into
the law of cosines. This gives 𝑎 squared is equal to
16 squared plus 10 squared minus two multiplied by 16 multiplied by 10 multiplied by
cos of 64 degrees. 16 squared is 256 and 10 squared is
100. So we have that 𝑎 squared is equal
to 356 minus 320 multiplied by cos of 64.
Now, we can evaluate this using a
calculator. But notice that the angle is given
in degrees which means we must make sure our calculator is in degree mode rather
than radians before we type this. Otherwise, we’ll get an incorrect
value for cos of 64 degrees. Typing this into my calculator, I
get that 𝑎 squared is equal to 215.72123.
Now, we’re nearly there. But I’m not asked to find the value
of 𝑎 squared. I ‘m asked to find the value of
𝑎. So I need to square root both
sides. Evaluating this square root on my
calculator gives 𝑎 is equal to 14.68745. Finally, the question asked me to
give the value of 𝑎 to two decimal places. So I need to round this value. To two decimal places, the value of
a is 14.69 centimeters.