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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
157
by the term sin2 x,
Also, if you divide both sides of
you obtain the identity
+
=
,
which you can simplify to:
1 + cot2 x = csc2 x.
This collection of related identities
(1) sin2 x + cos2 x = 1
(2) tan2 x + 1 = sec2 x
(3) 1 + cot2 x = csc2 x
is very important in simplifying some complex trigonometric expressions.
Be sure you can also recognize these
identities when they appear in other
forms:
1 – sin2 x = cos2 x
1 – cos2 x = sin2 x
sec2 x – 1 = tan2 x
csc2 x – 1 = cot2 x
Other important trigonometric identities include the addition formulas:
sin (A + B) = sin A cos B + sin B cos A
cos (A + B) = cos A cos B – sin A sin B
Since A – B = A + –B, you can also obtain
sin (A – B) = sin A cos (–B) + sin (–B) cos A = sin A cos B – sin B cos A
cos (A – B) = cos A cos (–B) – sin A sin (–B) = cos A cos B + sin A sin B
Letting A = B in the addition formulas, you can obtain the
identities:
sin 2A = 2 sin A cos A
cos 2A = cos2 A – sin2 A
double-angle
From this last double-angle identity you can derive the half-angle identities:
2
2
These identities can also be written:
sin2 x =
(1 – cos x)
cos2 x =
(1 + cos x)
2
cos 2A = cos A – (1 – cos A) = 2 cos A – 1
or,
cos2 A =
(1 + cos 2A)
cos 2A = (1 – sin2 A) – sin2 A = 1 – 2 sin2 A
or,
sin2 A =
(1 – cos 2A)
These are the identities that are most often applied in elementary analysis.
They will help you to simplify more complex trigonometric expressions and
to do some more difficult computations.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Find sin
. Since
is not a multiple of
or
or
that you should
be familiar with, you would not have been able to find this before, without
using your calculator. However, if you notice that
=
+
sin
=
+
= sin (
, you can rewrite
+
) = sin
=
cos
=
+
+ sin
.
+
a. cos
b. sin
c. cot
cos
+
=
Calculate:
Ans.:
a.
(
b.
(
c.
(
=
+ )
=
=
)
= .9659
Simplify cos (
– x).
cos (
– x) = cos
cos (–x) – sin
= (0) cos x – (1) (–sin x)
= sin x
To find tan
tan
, notice that
=
=
=
sin (–x)
Simplify:
a. cos (
–
)
b. tan (
+
)
.
=
Ans.:
a. –cos
b. –cot
=
=
= .4142
=
–
)
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
159
To obtain an identity for cos 3x, note 3x = 2x + x.
cos 3x = cos (2x + x) = cos 2x cos x – sin 2x sin x
= (cos2 x – sin2 x) cos x – 2 sin x cos x sin x
= cos3 x – sin2 x cos x – 2 sin2 x cos x
= cos3 x – 3 sin2 x cos x
= cos3 x – 3 (1 – cos2 x) cos x
= cos3 x – 3 cos x + 3 cos3 x
= 4 cos 3 x – 3 cos x
Right Triangle Trigonometry
Using the definitions of the trigonometric functions with the relationships of
the sides of a right triangle as described on page 148, you can calculate the
value of any trigonometric function defined on the angles of such a triangle.
If
is the angle in the triangle for which you wish to calculate the
trigonometric functions (
figure on page 148),
sin
is the angle placed at the center of the circle in the
=
tan
sec
,
=
cos
,
=
cot
,
csc
=
=
=
From these identities, you can calculate the trigonometric functions for any
angle by considering the angle to be placed in a right triangle at the center of
a circle. The reference circle is not necessarily the unit circle, since the
hypotenuse of the triangle is the radius. However, you can compute the
trigonometric function values by using the lengths of the sides of the triangle.
Write sin 3x using only the functions
sin x and cos x.
Ans.: 3 sin x cos2 x – sin3 x
or 3 sin x – 4 sin3 x
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
For example, if the hypotenuse is 5 and the length of the side opposite the
angle is 3, you can find the length of the third side (the side adjacent to
) by using the Pythagorean Theorem to obtain
=
=
= 4.
You can then calculate all the trigonometric functions of , without even
knowing the value of
:
sin
=
cos
=
tan
=
cot
=
sec
=
csc
=
Suppose that instead of knowing the lengths of the sides, you know that
tan
= . The opposite and adjacent sides are in the ratio of 2 : 3, or l2
: 3l. You can use the Pythagorean Theorem to find the hypotenuse,
=
=
=
. Since the values of the
trigonometric functions depend on the ratio of the sides, use 2 for the
opposite side, 3 for the adjacent side and
for the hypotenuse. The
values of the trigonometric functions are:
If sin
sin
=
cos
=
tan
=
cot
=
sec
=
csc
=
=
, then using x for the length of the opposite side and 2 for the
hypotenuse, the length of the adjacent side is
=
. The
trigonometric functions are:
sin
=
cos
=
tan
=
cot
=
sec
=
cot
=
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
161
The measurement unit used for angles is degree measure. When you want to
obtain the trigonometric functions for an angle of known degree measure,
move the angle to the center of the unit circle and look at the ratios of the
sides.
The central angle corresponding to an arc of length
around the unit
Use the diagrams on page 140.
circle is 30°. To find the trigonometric functions for this angle, use the
ratio of the sides of the right triangle with a 30° angle:
sin 30° =
cos 30° =
tan 30° =
cot 30° =
sec 30° =
csc 30° = 2
Angles that fit in the first quadrant are called acute angles. For larger angles
you use the unit circle to get the signs right the same way you do finding the
trigonometric functions for arcs. You can do this for angles measured either
in the positive (counterclockwise) or negative (counterclockwise) directions.
To find the trigonometric
functions for an angle of 135°, fit
the angle into the unit circle.
You can see that two angles are
formed in the first two quadrants,
the 135° angle, and the
supplementary 45° angle that
corresponds to the arc from the
end of the 135° angle to the point
(–1, 0).
Then calculate the trigonometric functions using the supplementary 45° angle
that lies in the second quadrant.
sin 135° =
cos 135° =
tan 135° = –1
cot 135° = –1
sec 135° =
csc 135° =
You can also use your calculator to find the value of the trigonometric functions
for angles other than integer multiples of 30° and 45°. Just be sure that you have
the calculator set in the appropriate mode.
1
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
To find sin 54°, put your calculator in degree mode and press the
key, then 54 and
. The calculator finds sin 54° = .8090.
Find:
a. cos 138°
b. cot 53°
c. sec
To get sec 54°, since there is no
key, you find cos 54° by using the
key. You have cos 54° = .5878. Then use the reciprocal key,
and
to find
To find tan
, make sure you are back in radian mode and use the
key, followed by
,
= 1.7013 = sec 54°.
(the 2nd function for the ^ key), ÷ , 15, and
. When you press
To obtain csc
and
, find sin
to obtain
you get tan
= .2126.
(.2079), then use the reciprocal key
= csc
= 4.8097.
Applications
Whenever you know the lengths of two sides of a right triangle you can find
the length of the third side using the Pythagorean Theorem and then the values
of all the trigonometric functions. If you have one of the acute angles and the
length of one side you can find all the remaining parts of the triangle since you
can find the other acute angle and then use the trigonometric functions to find
the other sides.
If you know that one angle of a right triangle is 26°, and the length of the
side opposite this angle is 11, you can find the hypotenuse from the
equation
sin 26° =
.
To find the length of the hypotenuse, (abbreviated to h here),
h sin 26° = 11
h=
h=
= 25.09
Ans.:
a. –.7431
b. .7536
c. 1.0642
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
163
You can find the adjacent side either by using the Pythagorean Theorem,
=
= 22.55, or,
tan 26° =
=
a tan 26° = 11
a=
=
= 22.55
You can apply this method for solving right triangles in many situations that
require you to find the lengths of parts of right triangles.
When you set the base of a ladder 5 feet from the base of a flat wall, the
ladder makes an angle of 48° with the ground. How high is the top of the
ladder?
The diagram on the right
shows the ladder leaning
against the wall, which
forms the side of the triangle
opposite the 48° angle.
The part of the triangle you wish to find is the side opposite the angle.
You already know the adjacent side, so the trigonometric function for you
to use is the tangent, since the tangent is the ratio of the lengths of the
opposite and adjacent sides.
How high would the ladder reach if
the angle with the ground was 73°
(and the bottom of the ladder was
still 5 feet from the base of the wall?
tan 48° =
1.1106 =
(1.1106)(5) = x
Ans.: 16.35 feet
x = 5.55 feet
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
The ladder reaches 5.55 feet up the wall. Since the ladder is 5 feet from
the base of the wall, you can use the Pythagorean Theorem to find the
length of the ladder.
=
=
= 7.47 feet
You are standing on the top of a cliff so that your eye level is 100 feet above
the surface of a lake, looking through a telescope at a boat that is approaching
the cliff. To keep the image of the boat in focus, you have to drop your
telescope 20° from the horizontal position. How far away from the cliff is the
boat?
The diagram on the right
shows the line of sight of the
telescope dropped 20° from
the horizontal line of sight to
the boat. Therefore the
complementary angle inside
the triangle is 70°.
You know that the side adjacent to the 70° angle is 100 feet long, and the
side opposite this angle is the quantity you are looking for. The
trigonometric function to use is again the tangent.
tan 70° =
2.7475 =
(2.7475)(100) = x
x = approximately 275 feet
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Suppose you walk due north at 4 miles per hour and continue for 3 hours.
Then you turn eastward 42° and continue at 3 miles per hour for 2 more hours
before you stop. How far are you from your starting point?
The diagram on the right is oriented
so that north is up. Notice that the
distance labeled y is opposite the 42°
angle in the top right triangle, and
the side labeled z is adjacent to the
42° angle. The hypotenuse of this
triangle is 6 miles since you traveled
for 2 hours at 3 miles per hour
during the second part of the trip.
The first (northern) leg of the trip
was 3 hours long at 4 miles per hour
and therefore covered a distance of
12 miles.
You can find the distance labeled y by using the sine of the 42° angle.
sin 42° =
.6691 =
(.6691)(6) = y
y = 4.01 miles
You can find the distance labeled z by using the cosine of the 42° angle.
cos 42° =
.7431 =
(.7431)(6) = z
z = 4.46 miles
Now you can find the straight-line distance, x, from start to end by using
the Pythagorean Theorem:
=
=
= 16.94 miles
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Exercises
1. Give the correct sign for each of the following:
a. sin 3
b. cos 7
c. tan 9
d. sin –5
e. sec –4
f. tan 3.5
2. Change to degree measure (each is given in radians):
a.
b.
c.
d.
e. 2
f.
3. Change to radian measure (each is given in degrees):
a. 120°
b. 540°
c. 225°
d. –150°
e. 144°
f. –444°
4. Find each of the following:
a. sin
b. cos
f.
g. sec
cot
c. sin
h. cot
d. tan
i.
5. Sketch graphs for each of the following functions
a. f(x) = 3 sin x
b. f(x) = 2 tan
c. f(x) = sec
x
x
d. f(x) = 4 cos 3x
e. f(x) = sin (2x – )
f. f(x) = 4 csc x – 3
g. f(x) = –2 cot (x +
)
h. f(x) = –1 + cos ( x –
i.
f(x) = 5 cos (2x – 1) – 2
j.
f(x) = –3 + 6 sec
)
x
6. Tell why each of the following is true or false (without calculation):
a. cos 2 > cos 3
b. sec (1 +
) = sec 1
c. tan 2 > sec 2
d. csc 9 is negative
e. sin 1 > sin (sin 1)
e. csc
csc
j. sin
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
7. Express each of the following in terms of sin x and cos x:
a. tan 2x
b. cos 4x
c. sin 4x
167
d. sin 3x + cos 2x
8. Use the trigonometric identities to find exact values for:
a. tan
b. cos
c. 2 sin
d. sec
e. cot
9. Find the lengths of all sides of the right triangle, given
a.
=
and the opposite side is 3.
b.
=
and the adjacent side is 5.
c.
= 23° and the hypotenuse is 7.
d. sin
=
.
e. tan
=
.
f.
cos
=
.
g. tan
=
h. sin
=
.
.
10.
If the angle of inclination from the ground to the top of a building is 72° at a point that is 50 feet from the
building, how high is the building?
11.
A woman is standing on a dock at some height above the water. She is holding one end of a rope attached to
the front end of a boat that is 8 feet from the dock. The rope is stretched taut and is making an angle of 58°
with her body. How high above the water is the dock?
12.
A lifeguard is on the edge of a beach looking straight out into the water. When he turns 65° to his right he sees
a drowning child 100 yards away from him in the direction he is facing. How far from the shore is the child?
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Answers
1. a. +
b. +
c. –
d. +
e. –
f. +
2. a. 300°
b. 315°
c. 48°
d. –13.37°
e. 114.59°
f. –42.97°
3. a.
b.
c.
d.
e.
f.
4. a.
b.
c.
d. –1
e. –1
f.
g.
h.
i.
j.
1
2
5. All the following graphs are in the same window: X: [–7, 7] Y: [–10, 10] with the x-axis tick marks (Xscl) set
at
and the y-axis tick marks (Yscl) at 1.
(a)
(b)
(d)
(c)
(e)
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
(f)
(g)
169
(h)
(i)
(j)
6. a. Both 2 radians and 3 radians are in the second quadrant, but 3 radians is closer to
. Since the cosine
becomes more negative as you move through the second quadrant, cos 3 is more negative than cos 2.
b. If sec (1 +
) = sec 1, then cos (1 +
) = cos 1. However, traveling 1 radian around the circle puts you
in the first quadrant, while a distance 1 +
radians around the circle leaves you in the second. The
cosine function is positive in the first quadrant, but negative in the second. Therefore, sec 1 and
sec (1 +
) have different signs and cannot be equal. The statement is false.
c. sin 2 < 1, since 2 radians is in the second quadrant. If you divide both sides of this inequality by the
negative number, cos 2, the direction of the inequality is reversed, so
tan 2 =
>
= sec 2
d. 9 radians is less than 3 radians, so 9 radians is in the second quadrant. The sine function, and therefore,
the cosecant function, is positive in the second quadrant. The statement is false.
e. Since sin 1 < 1, sin 1 radians is not as far around the first quadrant as 1 radian is. Therefore, the sine at 1
radian must be greater than the sine at sin 1 radians.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
b. 1 – 8 sin2 x + 8 sin4 x
7. a.
c. 4 sin x cos x – 8 sin3 x cos x
d. 1 + 3 sin x – 2 sin2 x – 4 sin3 x
8. a.
b.
c.
d.
9. a. hypotenuse = 6, adjacent side = 3
b. hypotenuse = 5
, opposite side = 5
c. opposite side = 7 sin 23° = 2.735, adjacent side = 7 cos 23° = 6.44
d. opposite side = 2, hypotenuse = 3, adjacent side =
e. opposite side = 7, adjacent side = 3, hypotenuse =
f.
adjacent side = x, hypotenuse = 5, opposite side =
g. opposite side = x, adjacent side = 1 + x, hypotenuse =
h. opposite side = 1, hypotenuse =
10.
11.
12.
50 tan 72° = 153.88 feet
= 5 feet
100 sin 25° = 42.26 yards
, adjacent side = x
e.