Video Transcript
Given the following figure, find
the lengths of 𝐴𝐶 and 𝐵𝐶 and the measure of angle 𝐵𝐴C in degrees. Give your answers to two decimal
places.
As the triangle is right angled, we
can use the trigonometrical ratios to work out the missing lengths and the missing
angles. The three ratios are sine, cosine,
and tangent, identified on your calculator by the sin, cos, and tan buttons. Sin 𝜃 is equal to the opposite
divided by the hypotenuse. Cos 𝜃 is equal to the adjacent
divided by the hypotenuse. And tan 𝜃 is equal to the opposite
side divided by the adjacent. Our first step is to label the
triangle. The side opposite the right angle,
the longest side, is called the hypotenuse. Side 𝐴𝐵 is the opposite, as it is
opposite the 22-degree angle. And side 𝐵𝐶 is adjacent or next
to the 22-degree angle.
Let’s first look at how we would
work out the length 𝐴𝐶, which we have labelled 𝑥 in this case. As we know the opposite, 𝐴𝐵, and
we are trying to work out the hypotenuse, 𝐴𝐶, we’re going to use the sine
ratio. Substituting in our values, gives
us sin 22 equals four divided by 𝑥. Multiplying both sides of this
equation by 𝑥, gives us 𝑥 multiplied by sin 22 equals four. And finally, dividing both sides by
sin 22 gives us 𝑥 equals four divided by sin 22.
Ensuring our calculator is in
degree mode, typing this into the calculator gives us 𝑥 equals 10.68 to two decimal
places. Therefore, the length 𝐴𝐶 in the
triangle is equal to 10.68. As we know now two sides of the
right-angled triangle, we could use Pythagoras’s theorem to calculate 𝐵𝐶. However, we’re going to stick with
the trigonometrical ratios. Length 𝐵𝐶 is the adjacent, length
𝐴𝐵, four, is the opposite. Therefore, we are going to use the
tan ratio. Tan 𝜃 equals the opposite divided
by the adjacent. If we let the length 𝐵𝐶 equal 𝑦,
our equation becomes tan 22 equals four divided by 𝑦.
Rearranging this equation in the
same way as before, multiplying by 𝑦, then dividing by tan 22, gives us 𝑦 equals
four divided by tan 22. Typing this into the calculator,
gives us a value for 𝑦 of 9.90. Therefore, the length of 𝐵𝐶 is
equal to 9.90. We now know all three lengths of
the triangle and two of the angles, 22 degrees and 90 degrees.
Our final step was to work out the
angle 𝐵𝐴𝐶, labeled 𝜃 on the diagram. Now we know that all angles in a
triangle add up to 180. Therefore, 90 degrees plus 22
degrees plus 𝜃 must equal 180 degrees. 90 add 22 is 112. So we have 112 plus 𝜃 equals
180. Subtracting 112 from both sides of
the equation, gives us a final answer of 𝜃 equals 68 degrees. Therefore, angle 𝐵𝐴𝐶 is equal to
68 degrees.
To summarise, we were asked to find
the lengths 𝐴𝐶 and 𝐵𝐶 as well as the angle 𝐵𝐴𝐶. We used the trigonometrical ratios
to calculate that 𝐴𝐶 was 10.68. And in a similar way, 𝐵𝐶 was
9.90. We then used the fact that angles
in a triangle add up to 180 degrees to work out that 𝐵𝐴𝐶 was equal to 68
degrees.
