Part a) State the exact value of
sin of 60 degrees.
Now, sin is one of our three
trigonometric ratios: sin, cos, and tan, which describe the ratio that exist between
the lengths of the sides of right-angled triangles with particular angles. Normally, when we’re using sin,
cos, or tan, we have access to a calculator, but here we don’t. And we’ve been asked to state the
exact value of sin of 60 degrees. This is because 60 degrees is a
special angle where we can write down the values of sin, cos, and tan exactly in
terms of surds or square roots.
There’re other special angles for
which we can do the same thing. And those angles are 30 degrees and
45 degrees. There’s a trick to help you
remember the values of sin and cos for these three angles. So we’ve written 30 degrees, 45
degrees, and 60 degrees across the top and sin and cos down the vertical side of
And first, what we do is we write
one, two, three across the top line for sin and then three, two, one across the
bottom line for cos. We then turn each of these values
into fractions with a denominator of two. So we have one over two, two over
two, and three over two in the top row for sin and three over two, two over two, and
one over two in the bottom row for cos.
Next, we include the square root of
each of the numerators — only the numerators, not the denominators. Now, the square root of one can
actually be simplified as the square root of one is actually just one because one
multiplied by one is equal to one. So sin of 30 degrees and cos of 60
degrees can just be simplified to one over two.
Now, that’s it! By remembering that method, we can
complete the table of the exact values of sin and cos for 30 degrees, 45 degrees,
and 60 degrees. We just need to write down the
value that we were asked for, which was sin of 60 degrees. Sin of 60 degrees is equal to root
three over two.
Part b) Using the diagram below,
work out the value of 𝑥.
Now, the diagram we’ve been given
is a diagram of a right-angled triangle, where one of the other angles is 60
degrees. So this suggests that we’re going
to be using what we’ve done in part a at some point in this question. In any case, we’re going to need to
apply some trigonometry to work out the value of 𝑥 as we’ve got a right-angled
triangle, in which we know one side, we know one other angle, and we want to
calculate the length of the second side.
For me, the first step with any
problem involving trigonometry is to write down the acronym SOHCAHTOA, which tells
us which of the trigonometric ratios sin, cos, and tan use which pairs of sides. So S, C, and T stand for sin, cos,
and tan and O, A, and H stand for opposite, adjacent, and hypotenuse, which are the
sides of this right-angled triangle.
Next, we’ll label the sides of this
triangle. So firstly, we have the hypotenuse
which is always the longest side and the side opposite the right angle. We also have the opposite which is
the side opposite the given angle of 60 degrees and finally the adjacent which is
the side between the right angle and the given angle of 60 degrees.
So the side that we know is the
adjacent. And the side that we’re looking to
work out is the hypotenuse. So A and H appear together in the
CAH part of SOHCAHTOA, which tells us that it is the cos ratio we’re going to be
using in this question.
The definition of the cos ratio is
that cos of an angle 𝜃 is equal to the length of the adjacent side divided by the
length of the hypotenuse. In our triangle, the angle 𝜃 is 60
degrees, the adjacent is four centimeters, and the hypotenuse which is what we’re
looking to calculate is 𝑥 centimeters. So we have the equation cos of 60
degrees is equal to four over 𝑥.
Now, we want to solve this equation
for 𝑥. But first of all, we need to know
what the value of cos of 60 degrees is. Well, looking back at the table
that we wrote down in part a, we can see that cos of 60 degrees is actually just
equal to the fraction one-half. So we can substitute this into our
equation. And it gives a half is equal to
four over 𝑥.
Now, 𝑥 is currently in the
denominator of the fraction on the right of this equation. So our next step in solving is
going to be to multiply both sides of the equation by 𝑥. When we multiply the left of the
equation by 𝑥, we have 𝑥 multiplied by a half which is equal to 𝑥 over two. And on the right, when we multiply
by 𝑥, this will cancel out the 𝑥 in the denominator. So we’re just left with four.
Now, we have 𝑥 over two or 𝑥
divided by two on the left of the equation. So we need to multiply both sides
of the equation by two. Multiplying 𝑥 over two by two will
just leave us with 𝑥 and multiplying four by two gives eight.
So we’ve answered the problem and
found that the value of 𝑥 is eight.