a brief review of geometry and trigonometry
DESCRIPTION
By Halliday ResnikTRANSCRIPT
GeometryIn geometry, the quantities that give the “size” of things include perimeter (the circumference is a circle’s perimeter—the distance around), area, and volume. Table 1 summarizes how to calculate these quantities for the most common geometrical shapes.
TrigonometryIn dealing with the size and arrangements of objects in space, you often need to figure out lengths and/or angles
that you don’t know from others that you do. The geometric relationships that exist in right triangles enable you to do this.
One useful relationship is that the ratio of correspond-ing sides is the same for all triangles having the same shape. By nesting the smaller triangles within the larger ones (Fig. 1a), we can see that all right triangles with the same acute angle q have the same shape. For a given q, we identify the three sides as the side adjacent to q (ADJ in Fig. 1b), the side opposite to q (OPP), and the hypotenuse (HYP), the longest side, which is opposite the right angle. The most commonly used ratios of sides are named the sine(sin), cosine (cos), and tangent (tan), and are defined as follows:
sinθ = OPPHYP (5)
cosθ = ADJHYP
(6)
tanθ = OPPADJ
(7)
(The inverses of these three ratios, in the same order, are called the cosecant (csc q), the secant (sec q) and the cotan-gent (cot q). These are much less commonly used.) Because their values depend on q, these ratios are functions of q, and are called the trigonometric functions. It is generally easi est to use your calculator to find the values of these functions, but you should know the values in Table 2 and the reasons for them.
A Brief Review of Geometry and Trigonometry
TABle 1 Useful Information about Geometric Shapes
Shape Figure Quantities
RECTANGLEb
w
Perimeter = 2w + 2h; Area = wh
TRIANGLE
h OR
b b
hArea = 1
2bh
CIRCLEr
Diameter D = 2r Circumference C = 2pr = pD Area =pr2
RECTANGU-LAR PRISM (box with all right angles)
b
l
w
Surface area = Sum of areas of six rectangular faces
Volume = (Area of base) × h = lwh
CYLINDER
h
r Volume = (Area of base) × h = 2prh
SPHERE
r
Surface area = 4pr2
Volume =43
3πr
HYP
ADJ
(a) (b)
q
qOPP
FiGuRe 1 Properties of Right Triangles. (a) All right triangles with the same acute angle have the same shape. (b) Identifying sides of a right triangle.
Physics Special market Book 2_Geometry and Trignometry.indd 1 2011-11-28 6:45:12 PM
A Brief Review of Geometry and Trigonometry2
The angles in a triangle always add up to 180°. The sum of the two acute angles in a right triangle must therefore be 90°. If one of these angles is q, the other must be 90°−q. A side that is opposite one of these two angles must be adjacent to the other, so
sin q = cos (90o – q) (8)
cos q = sin (90o – q) (9)
The Pythagorean theorem is a relationship between the sides of a right triangle:
Pythagorean
theoremOPP ADJ HYP2 2 2+ = (10)
If we divide both sides of Eq. 10 by HYP2, and then use the definitions in Eq. 5 and 6, we get another useful form of the Pythagorean theorem:
sin2 q + cos2 q = 1 (11)
If you know either the sine or the cosine of an angle, you can use this equation to find the other trigonometric function.
You can think of an angle as an amount of rotation—a total of 360° for each complete revolution. An angle q representing an amount of rotation can have any value up to infinity. We can also have negative values of q, represent-ing rotations in the opposite direction. When we define trig-onometric functions in terms of the sides of a right triangle (Eq. 5 to 7), the definitions are valid only for the angles that can actually occur in right triangles—those between 0 and 90°. The unit reference circle can be used to develop defini-tions of the sine and cosine that are valid for all possible values of q from −∞ To + ∞. For the sines and cosines of angles between 0° and 90°, they give the same values as Eq. 5 and 6.
Since you are back to facing in the same direction each time you rotate by 360°, the trigonometric functions have the same value each time you increase or decrease q by 360°. If an angle is not between 0 and 360°, you can always add an integer multiple of 360° to the angle to find an angle between 0 and 360° that has the same sine and cosine. In addition, for any angle between 0 and 360°, there are simple relationships that will let you find the trigonometric func-tions of the angle if you have values available for the angles in the first quadrant (those between 0 and 90°). These are summarized in Table 3.
The following additional formulas are sometimes useful when doing calculations involving the sides and angles of triangles. These formulas are good for any triangles, not just right triangles. In each formula, the side a (lower case) is oppo-site angle A (upper case) of the triangle, and so forth. Apart from maintaining this consistency, it doesn’t matter which letter you use for which angle and side.
Law of sinessin sin sinA
aB
bC
c= = (12)
Law of cosines c2 = a2 + b2 – 2ab cos C (13)
In the law of cosines, if the angle C = 90°, then the side c oppo-site it is the hypotenuse of a right triangle. Since cos 90° = 0, the law then says
HYP2 = c2 = a2 + b2
in other words, it reduces to the Pythagorean theorem.Below are some additional formulas which are sometimes
of value when working with trigonometric functions.
sin (q + f) = sin q cos f + cos q sin f (14)
sin (q – f) = sin q cos f – cos q sin f (15)
cos (q + f) = cos q cos f – cos q cos f (16)
cos (q – f) = cos q cos f + cos q cos f (17)
sin 2q = 2 sin q cos q (18)
cos 2q = cos2 q – sin2 q (19)
The last two formulas are simply Eq.14 and 16 rewritten for the special case when f = q.
Table 2 Numerical Values of Trigonometric Functions
Angle q in degrees
Angle q in radians sin q cos q tan q
0 0 0 1 0
90 p/2 1 0 ∞
180 p 0 –1 0
270 3p/2 −1 0 −∞
360 2p 0 1 0
Physics Special market Book 2_Geometry and Trignometry.indd 2 2011-11-28 6:45:13 PM
A Brief Review of Geometry and Trigonometry 3
TAB
le 3
W
hat T
o D
o W
hen
Ang
les
Are
bet
wee
n 90
° and
360
° (be
twee
n p/
2 an
d 2p
).
Qua
dran
t
Seco
nd q
uadr
ant:
9018
02
°≤≤
°≤
≤
θ
πθ
π
1
q
(–x,
y)
(x, y
)1
180°
–q
sin
q
sin
sin
sin(
)θ
θθ
==
°−y 1
180
cos
q
cos
cos
cos(
)θ
θθ
=−
=−
°−x 1
180
tan
q
tan
tan
tan(
)θ
θθ
=−
=−
°−y x
180
Thir
d qu
adra
nt:
180
270
3 2°≤
≤°
≤≤
θπ
θπ
1
q
(–x,
–y)
(x, y
)1
q–
180°
sin
sin
sin(
)θ
θθ
=−
=−
−°
y 118
0co
sco
sco
s()
θθ
θ=
−=
−−
°x 1
180
tan
tan
tan(
)θ
θθ
=− −
==
°−y x
y x18
0
Four
th q
uadr
ant:
270
360
3 22
°≤≤
°≤
≤
θ
πθ
π
1
(x, –
y)
(x, y
)1
q
360°
–q
sin
sin
sin(
)θ
θθ
=−
=−
°−y 1
360
cos
cos
cos(
)θ
θθ
==
°−x 1
360
tan
tan
tan(
)θ
θθ
=−
=−
°−y x
360
Physics Special market Book 2_Geometry and Trignometry.indd 3 2011-11-28 6:45:18 PM
Physics Special market Book 2_Geometry and Trignometry.indd 4 2011-11-28 6:45:18 PM