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

qq

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