Tangent and Right Triangles
The tangent function is the function defined as
tan(θ) =
sin(θ)
cos(θ)
The implied domain of the tangent function is every number θ except for
5π
those which have cos(θ) = 0 : the numbers . . . , − π2 , π2 , 3π
2 , 2 ,...
Graph of tangent
Period of tangent
Tangent is a periodic function with period π, meaning that
tan(θ + π) = tan(θ)
This follows from Lemma 10 of the previous chapter which stated that
cos(θ + π) = − cos(θ)
and
sin(θ + π) = − sin(θ)
Therefore,
tan(θ + π) =
sin(θ + π)
− sin(θ)
sin(θ)
=
=
= tan(θ)
cos(θ + π) − cos(θ) cos(θ)
236
Tangent is an odd function
Recall that an even function is a function f (x) that has the property that
f ( x) = f (x) for every value of x. Examples of even functions include x2 ,
x4 , x6 , and cos(x).
An odd function is a function g(x) that has the property g( x) = g(x)
for every value of x. Examples of odd functions include x3 , x5 , x7 , and sin(x).
We can add tangent to our list of odd functions. To see why tangent is
odd, we’ll use that sine is odd and cosine is even:
tan( ✓) =
*
*
*
*
sin( ✓)
sin(✓)
=
=
cos( ✓)
cos(✓)
*
*
*
*
/4*L
tan(✓)
* S* /
Co?€
*
*
*
Trigonometry for right triangles
Trigonometry for right triangles
Suppose that we have a right triangle. That is, a triangle one of whose
Suppose
a right
triangle.that
That
is, a triangle
one of whose
angles equals
call⇡ we
We that
the have
side of
the triangle
is opposite
the right
equals
call the side of the triangle that is opposite the right
2 . We
angle the angles
hypotenuse
of the
triangle.
angle the hypotenuse of the triangle.
.
kypoten u se.
If we focus our attention on a second angle of the right triangle, an angle
that we’ll call 8, then we can label the remaining two sides as either being
opposite from
8, orfocus
adjacent
to 8. We’ll
the lengths
sides of an angle
If we
our attention
on call
a second
angle of the
the three
right triangle,
the triangle
hyp,
opp,call
and✓, adj.
that
we’ll
then we can label the remaining two sides as either being
opposite from ✓, or adjacent to ✓. We’ll call the lengths of the three sides of
the triangle hyp, opp, and adj. See the picture on the following page.
237
We call the side of the angles
triangleequals
that is opposite
We call the
the right
side of the triangle that is opposite the right
se of the triangle.
angle the hypotenuse of the triangle.
.
hypot€nus€
oppOte.
GkcE. nt
aci
ttention on a second
angle
of
the
right
anon
angle
If we
focus
ourtriangle,
attention
a second
triangle, an angle
The following
proposition
relates
the
lengths
of theangle
sidesof
of the
the right triangle
hen we can label
the
remaining
two
sides
as
either
being
call 8, then
that
weangle
can label
thethe
remaining
two sides
as either being
shown above
to we’ll
the measure
of the
θ using
trigonometric
functions
adjacent to sine,
6. We’ll
call
the tangent.
lengths
of or
theadjacent
three sides
opposite
from 6,
to 6.of We’ll call the lengths of the three sides of
cosine,
and
(e)
(co),
pp, and adj.
the triangle hyp, opp, and adj.
Proposition (13). In a right triangle as shown above, the following equations hold:
sin(θ) =
opp
hyp
cos(θ) =
s(e)
adj
hyp
z
tan(θ) =
opp
adj
Proof: We begin with the triangle on the top right of this page, a right
CO (e)
triangle with its side lengths labelled as instructed above: hyp for the hypotenuse, opp for the side opposite θ, and adj for the remaining side, which
is adjacent to θ. The triangle below on the left is the first triangle scaled by
1
the number hyp
. That is, all lengths have been divided by the number hyp,
but the angles remain unchanged. What we have then is a new right triangle
whose hypotenuse has length hyp
hyp = 1.
e three formulas
Then we have the three formulas
=
opp
hyp
—
cos(6)
=
adj
hyp
—
opp
tan(6)
sin(6)==
adj
hyp
cos(6)
—
kyp
216
=
adj
hyp
—
SIi
tan(6)
(e))
=
216
a4j
kyp
The picture on the right shows sin(θ) and cos(θ). Notice that the triangles
opp
adj
from the left and right are the same, so sin(θ) = hyp
and cos(θ) = hyp
.
238
Last, notice that from the definition of tangent we have
opp
hyp
adj
hyp
sin(θ)
tan(θ) =
=
cos(θ)
=
opp
adj
Problem. Find sin(8), eos(8), and tan(8) for the angle 8 shown belo
Problem. Find sin(θ), cos(θ), and tan(θ) for the angle θ shown below.
5
5
3
3
It
Lj.
Solution. The hypotenuse is the side that is opposite the right angle
Solution. The hypotenuse is the
side that is opposite the right angle. It has
length 5. Of the two remaining sides, the one that is opposite of 8 ha
length 5. Of the two remaining sides, the one that is opposite of θ has length
4, and the one that is adjacent to 0 has length 3. Therefore,
4, and the one that is adjacent to θ has length
3. Therefore,
O]3J3
4
4 sin(0)
opp
=
sin(θ) =
hyp 5
—
—
--
*
*
*
*
adj
3
cos(θ) =
=
hyp 5
cos(8)
opp 4
tan(θ) =
=
adj
3
tan(0)
*
*
*
239
*
—
adj
—
3
--
—
—
4
--
*
*
*
*
*
Exercises
sin(✓)
For #1-7, use the definition of tangent, that tan(✓) = cos(✓)
, to identify the
given value. You can use the chart on page 227 for help.
1.) tan
⇡
3
5.) tan
⇡
6
2.) tan
⇡
4
6.) tan
⇡
4
3.) tan
⇡
6
7.) tan
⇡
3
4.) tan(0)
Suppose that is a real number, that 0   ⇡2 , and that cos( ) = 13 .
Use Lemmas 7-12 from the previous chapter, the definition of tangent, and
the periods of sine, cosine, and tangent to find the following values.
8.) sin( )
14.) cos(
)
9.) tan( )
15.) sin(
)
10.) sin( + ⇡2 )
16.) cos( + 2⇡)
⇡
2)
17.) sin( + 2⇡)
11.) cos(
12.) cos( + ⇡)
18.) tan( + ⇡)
13.) sin( + ⇡)
240
Match the numbered piecewise defined functions with their lettered graphs
below.
(
sin(x)
19.) f (x) =
tan(x)
if x ∈ [0, ∞); and
if x ∈ (− π2 , 0).
(
tan(x)
20.) g(x) =
cos(x)
if x ∈ (0, π2 ); and
if x ∈ (−∞, 0].
(
tan(x)
21.) h(x) =
cos(x)
if x ∈ [0, π2 ); and
if x ∈ (−∞, 0).
(
sin(x)
22.) p(x) =
tan(x)
if x ∈ (0, ∞); and
if x ∈ (− π2 , 0].
A.)
/2
B.)
C.)
241
Exercises
Exercises
Exercises
Exercises
cos(6), and Exercises
tan(6) for the angles
Exercises
Find sin(6),
6 given below. You might
might
Find Find
sin(6),
cos(6),
and
tan(6)
for the
6 given
You
theorem
tobelow.
findbelow.
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using
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might
sin(6),
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forforangles
the
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sin(6),
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and
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theangles
angles
given
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to
find
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cos(θ),
and
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for
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θ
given
Find
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and
fortheorem
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