Download Basic Trigonometry Review

Basic Review of Trigonometry
RIGHT TRIANGLE
The 6 basic trigonometric functions:
hyp
1) sin = opp / hyp
2) cos = adj / hyp
3) tan = opp / adj
opp
4) csc = hyp / opp
5) sec = hyp / adj
6) cot = adj / opp
adj
SPECIAL TRIANGLES
1
sin 30 = 2
30 -60 -90
45 -45 -90
2
3
2x
cos 30 = 2
60
x
sin 60 =
30
2x
3
2
x
2
cos 45 = 2
1
cos 60 = 2
3x
sin 45 = 2
45
45
x
Complementary Angles Rules:
If and are complementary angles (add up to 90 ), then sin = cos and cos
Another way of stating this is: sin (90 - ) = cos and cos (90 - ) = sin .
= sin .
UNIT CIRCLE
Given a circle of radius 1, centered at the origin on an x-y coordinate system. The equation of this
circle is x2 + y2 = 1. If we let be an angle with initial side along the positive x-axis and terminal side a
radius of the circle passing through the point P(x, y), then cos = x and sin = y.
y
The basic trigonometric functions are:
sin = y
cos = x
tan = y/x
csc = 1/y
sec = 1/x
cot = x/y
(0,
1)
r=1
(-1,
0)
x
can be measured in degrees or radians
1. 180 =
(0,
-1)
2. Arc-length of a unit circle: s =
3. Arc-length of a circle of radius r : s = r
4. Area of sector of unit circle: A =
/2
5. Area of a sector of a circle of radius r : A =
1
2
r2
P (x,
y)
y
(1, x
0)
SINE, COSINE, and TANGENT of some basic angles:
Sine
Cosine
Tangent
0
1
0
1
2
3
2
3
3
rad.
2
2
2
2
1
rad.
3
2
1
2
rad.
1
0
undefined
rad.
3
2
1
-2
- 3
rad.
2
2
rad.
1
2
Angle
0 / 0 radians
30 / 6 rad.
45 /
4
60 /
3
90 /
2
2
3
3
4
5
6
120 /
135 /
150 /
-
2
2
-
3
2
3
-1
-
3
3
0
-1
0
rad.
1
-2
-
3
2
3
3
rad.
-
2
2
-
2
2
1
-
3
2
1
-2
rad.
-1
0
undefined
rad.
-
3
2
1
2
- 3
rad.
-
2
2
2
2
-1
rad.
1
-2
3
2
360 / 2 rad.
0
1
180 /
rad.
7
6
5
4
4
3
3
2
5
3
7
4
11
6
210 /
225 /
240 /
270 /
300 /
315 /
330 /
rad.
3
-
3
3
0
90 (0, 1)
1
2
(
(
2
2
3
2
1
2
,
3
2
) 120
2
, 2 ) 135
60 ( 12 ,
/2
2 /3
)
45 (
/3
3 /4
3
2
2
2
,
2
2
)
3
2
, 12 )
/4
(
30 (
5 /6
, ) 150
/6
(-1, 0) 180
0 or 2
7 /6
(
3
2
,
(
1
2
) 210
2
2
,
2
2
7 /4
4 /3
315
5 /3
(
1
2
,
3
2
) 240
3
2
11 /6 330
5 /4
) 225
0 or 360 (1, 0)
3 /2
300
1
2
,
2
2
,
,
1
2
2
2
3
2
270 (0, -1)
LSC-Montgomery Learning Center: Basic Review of Trigonometry
Last Updated April 13, 2011
Page 2
BASIC TRIGONOMETRIC IDENTITIES
1. Definitions
6. Product-to-Sum Formulas
tan
sin
cos
cot
cos
sin
sec
1
cos
csc
1
sin
sin A cos B = ½ [sin (A + B) + sin (A – B)]
cos A sin B = ½ [sin (A + B) – sin (A – B)]
cos A cos B = ½ [cos (A + B) + cos (A – B)]
sin A sin B = ½ [cos (A – B) – cos (A + B)]
7. Sum-to-Product Formulas
2. Pythagorean Identities
sin2 + cos2 = 1
tan2 + 1 = sec2
cot2 + 1 = csc2
sin A + sin B = 2sin
A+B
A-B
cos
2
2
sin A - sin B = 2cos
A+B
A-B
sin
2
2
3. Addition/Subtraction Formulas
cos A + cos B = 2cos
A+B
A-B
cos
2
2
sin (A + B) = sin A cos B + cos A sin B
sin (A – B) = sin A cos B – cos A sin B
cos (A + B) = cos A cos B – sin A sin B
cos (A – B) = cos A cos B + sin A sin B
tan A tan B
tan (A + B) =
1 tan A tan B
tan A tan B
tan (A – B) =
1 tan A tan B
8. Opposite Angle Formulas
4. Double-Angle Formulas
9. Reduction Formulas
sin 2 = 2 sin cos
cos 2 = cos2 – sin2
= 1 – 2 sin2
= 2 cos2 – 1
2 tan
tan 2 =
1 - tan2
1 cos 2
cos 2
2
1
cos
2
sin 2
2
sin ( + 2k ) = sin
cos ( + 2k ) = cos
tan ( + 2k ) = tan
cos A - cos B = (-2)sin
sin ( - ) = - sin
cos ( - ) = cos
tan ( - ) = - tan
2
=
sin ( /2 - ) = cos
cos ( /2 - ) = sin
tan ( /2 - ) = cot
cot ( /2 - ) = tan
sec ( /2 - ) = csc
csc ( /2 - ) = sec
11. The Law of Sines
sin A
a
1 - cos
2
sin B
b
sin C
c
or
a
sin A
b
sin B
c
sin C
12. The Law of Cosines
1 + cos
2
2
1 - cos
tan =
2
sin
cos
cot ( - ) = - cot
sec ( - ) = sec
csc ( - ) = - csc
10. Complementary Angle Formulas
5. Half-Angle Formulas
sin
A+B
A-B
sin
2
2
=
cos A
C
a
b
A
c
LSC-Montgomery Learning Center: Basic Review of Trigonometry
Last Updated April 13, 2011
cos B
B
cos C
b2
c2 a2
2bc
a2
c2 b2
2ac
a2
b2 c 2
2ab
or equivalently,
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
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