3.5 DoubleAngleIdentities - All-in-One High School
3.5. Double Angle Identities
3.5 Double Angle Identities
1.
If sin x =
4 5
and
in Quadrant
II, then
cosine and tangent are
negative.
Also, by the
Pythagorean Theorem, the
third side is 3(b =
52 - 42).
So,
cos x
=
-
3 5
and
tan x
=
-
4 3
.
Using this, we can find sin 2x, cos 2x, and
tan 2x.
sin 2x = 2 sin x cos x
= 2? 4 ?-3 55 24
=- 25
cos 2x = 1 - sin2 x
42 = 1-2?
5
= 1 - 2 ? 16 25
= 1 - 32 25
7 =-
25
2 tan x tan 2x = 1 - tan2 x
=
2
?
-
4 3
1-
-
4 3
2
=
-
8 3
1
-
16 9
= -8 3
?-7 9
= -8 ?-9 37
24 =
7
2. This is one of the forms for cos 2x.
cos2 15 - sin2 15 = cos(15 ? 2) = cos 30 3 = 2
3. Step 1: Use the cosine sum formula
cos 3 = 4 cos3 - 3 cos cos(2 + ) = cos 2 cos - sin 2 sin
Step 2: Use double angle formulas for cos 2 and sin 2 = (2 cos2 - 1) cos - (2 sin cos ) sin
Step 3: Distribute and simplify.
= 2 cos3 - cos - 2 sin2 cos = - cos (-2 cos2 + 2 sin2 + 1) = - cos [-2 cos2 + 2(1 - cos2 ) + 1] = - cos [-2 cos2 + 2 - 2 cos2 + 1] = - cos (-4 cos2 + 3) = 4 cos3 - 3 cos
Substitute 1 - cos2 for sin2
4. Step 1: Expand sin 2t using the double angle formula.
sin 2t - tant = tant cos 2t 2 sint cost - tant = tant cos 2t
50
Chapter 3. Trigonometric Identities and Equations, Solution Key
Step 2: change tant and find a common denominator.
sin t 2 sint cost -
cos t 2 sint cos2 t - sint
cos t sint(2 cos2 t - 1)
cos t sint ? (2 cos2 t - 1) cos t
tant cos 2t
5.
If
sin
x
=
-
9 41
and
in
Quadrant
III,
then
cos
x
=
-
40 41
and
tan
x
=
9 40
(Pythagorean Theorem, b =
412 - (-9)2).
So,
sin 2x = 2 sin x cos x 9 40
=2?- ?- 41 41
720 =
1681 6. Step 1: Expand sin 2x
cos 2x = 2 cos2 x - 1 40 2
=2 - -1 41
3200 1681 =-
1681 1681 1519 = 1681
sin 2x tan 2x =
cos 2x
720
=
1681 1519
1681
720 =
1519
sin 2x + sin x = 0 2 sin x cos x + sin x = 0
sin x(2 cos x + 1) = 0
Step 2: Separate and solve each for x.
sin x = 0
x = 0,
or
7. Expand cos 2x and simplify
2 cos x + 1 = 0
1 cos x = -
2 x = 2 , 4
33
cos2 x - cos 2x = 0 cos2 x - (2 cos2 x - 1) = 0
- cos2 x + 1 = 0 cos2 x = 1 cos x = ?1
cos x = 1 when x = 0, and cos x = -1 when x = . Therefore, the solutions are x = 0, . 8. a. 3.429 b. 0.960 c. 0.280
51
3.5. Double Angle Identities
9. a.
2 2 csc x 2x =
sin 2x 2
2 csc x 2x = 2 sin x cos x 1
2 csc x 2x = sin x cos x
2 csc x 2x = sin x
1
sin x sin x cos x
sin x 2 csc x 2x = sin2 x cos x
1 sin x 2 csc x 2x = sin2 x ? cos x 2 csc x 2x = csc2 x tan x
b.
cos4 - sin4 = (cos2 + sin2 )(cos2 - sin2 ) cos4 - sin4 = 1(cos2 - sin2 ) cos 2 = cos2 - sin2
cos4 - sin4 = cos 2
c. 10. cos 2x - 1 = sin2 x
sin 2x
2 sin x cos x
1 + cos 2x = 1 + (1 - 2 sin2 x)
sin 2x 2 sin x cos x 1 + cos 2x = 2 - 2 sin2 x
sin 2x 2 sin x cos x 1 + cos 2x = 2(1 - sin2 x)
sin 2x 2 sin x cos x
= 1 + cos 2x
2 cos2 x
sin 2x sin x =
1 + cos 2x cos x
sin 2x = tan x
1 + cos 2x
(1 - 2 sin2 x) - 1 = sin2 x -2 sin2 x = sin2 x 0 = 3 sin2 x 0 = sin2 x 0 = sin x x = 0,
52
Chapter 3. Trigonometric Identities and Equations, Solution Key
11.
cos 2x = cos x 2 cos2 x - 1 = cos x 2 cos2 x - cos x - 1 = 0 (2 cos x + 1)(cos x - 1) = 0
2 cos x + 1 = 0 or cos x - 1 = 0
2 cos x = -1
cos x = 1
cos x = - 1 2
cos x
=
1
when
x
=
0
and
cos x
=
-
1 2
when
x
=
2 3
.
12.
13. sin 2x - cos 2x = 1
2 csc 2x tan x = sec2 x
2 sin x 1 sin 2x ? cos x = cos2 x
2
sin x 1
2 sin x cos x ? cos x = cos2 x
1
1
cos2 x = cos2 x
2 sin x cos x - (1 - 2 sin2 x) = 1 2 sin x cos x - 1 + 2 sin2 x = 1 2 sin x cos x + 2 sin2 x = 2 sin x cos x + sin2 x = 1 sin x cos x = 1 - sin2 x sin x cos x = cos2 x
? 1 - cos2 x cos x = cos2 x
1 - cos2 x cos2 x = cos4 x cos2 x - cos4 x = cos4 x
cos2 x - 2 cos4 x = 0 cos2 x(1 - 2 cos2 x) = 0
1 - 2 cos2 x = 0
cos2 x = 0
- 2 cos2 x = -1
cos x = 0 or x = , 3 22
cos2 x = 1 2 2
cos x = ? 2
x = , 5 44
Note: If we go back to the equation sin x cos x = cos2 x, we can see that sin x cos x must be positive or zero, since cos2 x is always positive or zero. For this reason, sin x and cos x must have the same sign (or one of them
53
3.5. Double Angle Identities
must
be
zero),
which
means
that
x
cannot
be
in
the
second
or
fourth
quadrants.
This
is
why
3 4
and
7 4
are
not
valid solutions.
14. Use the double angle identity for cos 2x.
sin2 x - 2 = cos 2x sin2 x - 2 = cos 2x sin2 x - 2 = 1 - 2 sin2 x
3 sin2 x = 3 sin2 x = 1 sin x = ?1 x = , 3 22
54
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