𝐴𝐵𝐶 is a triangle, where 𝑎 is
equal to nine, 𝑏 is equal to six, and the measure of the angle at 𝐴 is equal to
58.1 degrees. Find the measure of the angle at 𝐵
to the nearest tenth of a degree.
Let’s start by sketching this
diagram out. Remember the diagram does not need
to be to scale. But it should be roughly in
proportion. So we can check the suitability of
any answers we get. The measure of the angle at 𝐴 is
58.1 degrees. Remember the side opposite the
angle labelled 𝐴 is lowercase 𝑎. So that one is nine. Similarly, the side opposite the
angle labelled as 𝐵 is lowercase 𝑏. That’s six.
So we have a non-right-angled
triangle, in which we know two sides and the measure of one angle. We need to decide then whether we
use the law of cosines or the law of sines. The law of cosines is used when we
either know or are trying to calculate the size of the included angle.
In our diagram, the included angle
is 𝐶. It’s the one that lies directly
between the two sides given. Since we don’t know the size of
that angle, we need to use the law of sines: sin 𝐴 over 𝑎 is equal to sin 𝐵 over
𝑏 which is equal to sin 𝐶 over 𝑐. Another thing to look out for our
pairs of sides and their associated angles, that is the given angles must sit
directly opposite the given side as in this triangle.
With the law of sines, we only need
to use two parts of the equation. Here, we know the length of the
side 𝑎 and the angle 𝐴. And we know the length of side 𝑏
and we’re trying to calculate the measure of the angle at 𝐵. We use sin 𝐴 over 𝑎 is equal to
sin 𝐵 over 𝑏. Let’s substitute what we know into
Remember we can use this formula
either way round. When calculating the angle, it’s
sensible to have the sin part of the formula on the top to minimise any rearranging
required to solve the equation. In this case, the equation becomes
sin of 58.1 degrees over nine is equal to sin of 𝐵 degrees divided by six.
We can solve this equation by
multiplying both sides by six. sin 𝐵 is equal to sin of 58.1 degrees divided by
nine then multiplied by six. Popping that into our calculator
gives us that sin of 𝐵 is equal to 0.5659.
Next, we’ll calculate the inverse
sine of both sides of this equation. We do this because the inverse sine
of sin 𝐵 is simply 𝐵. We’ll use the exact value we
calculated before because this will minimise any errors in calculations from
rounding too early. 𝐵 is equal to the inverse sin of
0.5659. This gives us that the measure of
the angle at 𝐵 is 34.470.
Correct to the nearest tenth of a
degree, the measure of the angle at 𝐵 is 34.5 degrees.