Download The Principal Trigonometric Ratios (Right Triangles)

The Mathematics 11
Competency Test
The Principal Trigonometric
Ratios (Right Triangles)
This document begins our brief discussion of trigonometry (from the Greek words for “measuring
triangles”). The initial goal of trigonometric methods is to use measures of some sides and/or
angles of a triangle to calculate the measures of ALL other unknown sides and/or angles of that
triangle. This process is called solving the triangle.
For the BCIT Mathematics 11 Competency Test, you
B
only need to be able to solve right triangles. However,
the methods described briefly here can be extended to
plane triangles which do not contain a right angle.
c
Right triangle trigonometry is based on the use of ratios
of lengths of various pairs of sides of the right triangle.
The three most useful ratios are:
A
sine of angle A
≡ sin A ≡
=
b
a
C
length of leg opposite angle A
length of hypotenuse
a
c
cosine of angle A
≡ cos A ≡
=
length of leg adjacent to angle A
length of hypotenuse
b
c
and
tangent of angle A
≡ tan A ≡
=
length of leg opposite angle A
length of leg adjacent to angle A
a
b
Any two right triangles that have the same angle A will be similar, since they have the angle A in
common, and the right angle in common. Therefore, using the properties of similar triangles, the
values of these ratios
sin A, cos A, tan A
David W. Sabo (2003)
The Principal Trigonometric Ratios (Right Triangles)
Page 1 of 6
will be the same for both of those triangles (and indeed, for any right triangle having the same
angle A). This observation means that the values of the sine, cosine and tangent ratios depend
only on the measure of the angle to which they refer.
The importance of this observation is that if we have an independent way of getting values for
sin A, cos A, and tan A, from just knowing the value of the angle A, and we know the length of at
least one side of a right triangle, then we can use the definitions of the appropriate ones of these
ratios as given above to determine the lengths of the remaining sides of the right triangle.
(Using methods developed in more advanced branches of mathematics, formulas for sin A,
cos A, and tan A in terms of the angle A have been developed. These formulas are programmed
into scientific calculators, and are accessible using keys normally labelled by the ratio names:
sin, cos, and tan.)
B
Example 1: Solve the right triangle shown in the
figure to the right.
solution:
Here we know the following properties of the right
triangle:
c
a
410
A = 410, C = 900, and b = 17
To “solve” this triangle means we need to determine
values for the remaining unknown parts – in this
case, the values of B, a, and c.
C
b = 17
A
Since for a right triangle,
A + B = 900,
we get immediately that
410 + B = 900
so that
B = 900 – 410 = 490
That was the easy part.
Now, to determine the length ‘a’, we need to find a trigonometric ratio from the previous list which
involves ‘a’ (the unknown length to be determined) and ‘b’ (a length we already know). Checking
the list, we see that the tangent ratio satisfies this requirement, giving
tan A =
a
b
or
tan 410 =
a
17
so that
David W. Sabo (2003)
The Principal Trigonometric Ratios (Right Triangles)
Page 2 of 6
(
)
a = (17 ) tan 410 .
Now, using your scientific calculator (make sure it is in “degree” mode), you can easily determine
that
tan 410 ≅ 0.8693
(we’ve rounded to four decimal places – your calculator probably gives you at least eight or nine
decimal places), so that
a = (17)(tan 410) ≅ (17)(0.8693) = 14.7781
To determine the length c, we could now use Pythagoras’s Theorem (since we now know the
lengths of the other two sides of the right triangle) or we could use the fact that
cos A =
b
c
cos 410 =
⇒
17
c
which gives
c=
17
17
≅
≅ 22.5255
0
0.7547
cos 41
Thus, the requested “solution” here is
B = 490,
a ≅ 14.7781
and
c ≅ 22.5255
(By the way, notice that
a2 + b2 ≅ 14.77812 + 172 ≅ 507.3922
and
c2 ≅ 22.52552 ≅ 507.3982
which, except for a slight discrepancy due to our rounding of the values of a and c, indicates that
our solution satisfies Pythagoras’s Theorem – which is must, of course.)
When we do calculations involving values of trigonometric ratios (such as the calculation of a and
c in Example 1 above, we normally do not write down rounded-off values of the trigonometric
ratios as intermediate results. We did that in the work above to help you follow the details of the
calculation. Generally, all calculations are done to the full precision of your calculator, and only
final results are rounded-off. By using bracket keys and ordering the keystrokes appropriately, it
is not necessary to manually record and re-enter any intermediate results in these calculations,
eliminating one needless source of error in these calculations. If you do not round off until the
final answer in each case, you will get slightly different answers than above: a ≅ 14.7779 and
c ≅ 22.5252, rounded to four decimal places. For a detailed discussion of the implications of
these differences in answers, you can read the documents on this website on the topic of
approximate and exact numbers.
David W. Sabo (2003)
The Principal Trigonometric Ratios (Right Triangles)
Page 3 of 6
Remarks:
• Everything we are doing here applies in the ways illustrated ONLY to right triangles.
• the ratios sine, cosine, and tangent, are associated with each of the two acute angles in the
right triangle, using the abbreviated definitions:
sine
cosine
that is:
s↔
O
H
Adjacent
Hypotenuse
that is:
c↔
A
H
Opposite
Adjacent
that is:
t↔
O
A
Opposite
Hypotenuse
↔
↔
and
tangent
↔
Here “Opposite” refers to the length of the side opposite the angle in question, and “adjacent”
refers to the leg of the triangle adjacent to the angle in question.
A mnemonic device many find useful in recalling these associations strings the bolded first
characters above into
“SOH-CAH-TOA”
(Sound these otherwise nonsense words out a few times and write them out before attempting to
solve a right triangle trigonometric problem, and they may help you recall that the sine of an angle
is the ratio of the side opposite to the hypotenuse, the cosine of an angle is the ratio of the side
adjacent to the hypotenuse, and the tangent of an angle is the ratio of the side opposite to the
side adjacent.)
Our original definitions of sine, cosine, and tangent, at the beginning of this document gave
formulas only with respect to angle A in the right triangle. The definitions above apply to both
angle A and angle B (and, in fact, with a bit of massaging, even to angle C, though that is not a
very useful thing here). Make sure that you can demonstrate how the formulas for sin A, cos A,
and tan A given on the first page of this document are obtained from the definitions above.
• Mathematicians have various theories about where the names sine, cosine, and tangent,
came from. You can read about some of these ideas on the internet, if you are interested, but
they probably won’t help you much in learning how to do trigonometry.
• Trigonometry books often define three more trigonometric ratios which are reciprocals of the
three trigonometric ratios defined above. They are
cosecant A = csc A =
secant A = sec A =
1
sin A
1
cos A
and
cotangent A = cot A =
David W. Sabo (2003)
1
tan A
The Principal Trigonometric Ratios (Right Triangles)
Page 4 of 6
The availability of values for these three ratios was useful in pre-calculator days in helping people
do trigonometric calculations to avoid having to do division arithmetic by hand. When it was
necessary to divide by sin A, for instance, one could simply multiply by csc A. Since hand-held
electronic calculators do division as easily as multiplication, these three additional ratios are no
longer as useful as they once were. You’ll notice that your calculator does not have function keys
for these three ratios. (However, if you absolutely need values of any of these three ratios, you
can get them by using the sine, cosine, or tangent function keys, followed by the reciprocal or 1/x
key.)
• As a more general remark, notice that in Example 1, we needed to refer to all three sides (using
lower case letters: a, b, and c) and all three angles (using upper case letters: A, B, and C) during
the course of the calculations. This means that it is absolutely essential to distinguish between
upper case and lower case letters as symbols in these calculations. You will only cause needless
confusion if sometimes A refers to an angle and sometimes, in the same problem, A refers to the
length of the side opposite angle A. This distinction between upper and lower case symbols in
mathematics should always be carefully observed.
Before ending this introduction to the principal trigonometric ratios (sine, cosine and tangent),
we’ll give you a couple of brief examples that you can use to practice your mastery of their
definitions.
B
Example 2: Write down the values of the three
principal trigonometric ratios for angle B in the right
triangle sketched to the right.
solution:
sin B =
opposite
120
=
hypotenuse 136
( ≅ 0.8824 )
cos B =
adjacent
64
=
hypotenuse 136
( ≅ 0.4706 )
tan B =
opposite 120
=
adjacent 64
136
64
C
A
120
( = 1.875 )
Example 3: Determine the exact values of the
principal trigonometric ratios for the angle θ shown in
the right triangle in the sketch.
11
solution:
This is a right triangle, so the definitions of sine,
cosine, and tangent given in this document can be
used. To be able to calculate the required ratios, we
will need the length of the hypotenuse. Using
Pythagoras’s Theorem, we get
19
θ
c 2 = a2 + b2
David W. Sabo (2003)
The Principal Trigonometric Ratios (Right Triangles)
Page 5 of 6
= 192 + 112 = 482
Since we are to write down expressions for the exact values for the three ratios, we will use
c=
482
without working out a decimal approximation for this square root.
So, the completed triangle is as shown in the
sketch to the right. Even though we don’t have
any formulas for sine, cosine, or tangent referring
to an angle θ (a Greek symbol commonly used in
mathematics to indicate an angle), from the sketch
it is still clear which sides are “opposite,”
“adjacent,” and the “hypotenuse” in this triangle.
We get
sinθ =
opposite
=
hypotenuse
cos θ =
adjacent
=
hypotenuse
tanθ =
opposite 11
=
adjacent 19
11
19
482
θ
11
482
19
482
and
David W. Sabo (2003)
The Principal Trigonometric Ratios (Right Triangles)
Page 6 of 6