## SAT Physics Subject Test

## Chapter 1 Math Review

The few questions on the SAT Physics Subject Test that require you to know mathematics are straightforward and actually need little math beyond some algebra and maybe a little trig. In this chapter we are going to help you brush up on some knowledge you probably don”t use every day, such as trig and the properties of vectors and how they are used. The material in this chapter is pretty clear-cut, so you should know this stuff backward and forward for the test.

**SCIENTIFIC NOTATION**

It”s usually much easier to write very large or very small numbers in scientific notation. For example, the speed of light through empty space is approximately 300,000,000 meters per second. In scientific notation, this number would be written as 3 × 10^{8}. Here”s another example: In standardunits, Newton”s universal gravitational constant is about 0.0000000000667; in scientific notation, this number would be written as 6.67 × 10^{–11}. In general, we say that a number is in **scientific notation** when it”s written in the form *a* × 10* ^{n}*, where 1 ≤

*a*< 10 and

*n*is an integer. As the two examples above show, when a very large number is written in scientific notation, the value of

*n*is a large positive integer, and when a very small number is written in scientific notation,

*n*is a negative integer with a large magnitude. To multiply or divide two numbers written in scientific notation, just remember that 10

*× 10*

^{m}*= 10*

^{n}*and 10*

^{m + n}*/10*

^{m}*= 10*

^{n}

^{m}^{–}

*. So, for example, (3 × 10*

^{n}^{8})(2.5 × 10

^{–12}) = 7.5 × 10

^{–4}and (8 × 10

^{9})/(2 × 10

^{–5}) = 4 × 10

^{14}.

### BASIC TRIG REVIEW

If you”re given a right triangle, there are certain special functions, called **trig functions**, of the angles in the triangle that depend on the lengths of the sides. We”ll concentrate on three of these functions; the **sine**, **cosine**, and **tangent** (abbreviated sin, cos, and tan, respectively). Take a look at the following right triangle, *ABC*. The right angle is at *C*, and the lengths of the sides are labeled *a*, *b*, and *c*.

First, we”ll mention one of the most important facts about any right triangle. The **Pythagorean theorem** tells us that the square of the *hypotenuse* (which is the name of the side opposite the right angle, always the longest side) is equal to the sum of the squares of the other two sides (called the *legs*):

*a*^{2} + *b*^{2} = *c* ^{2}

**Triangles and Beans?**

Pythagoras was

actually a cult leader

around 500 B.C. Some rules

of his number-worshipping

group included prohibitions

against eating beans and

wearing wool.

Now for the trig functions. Let”s consider angle *A* in the right triangle pictured above. The sine, cosine, and tangent of this angle are defined like this:

By *opposite* we mean the length of the side that”s opposite the angle, and by *adjacent* we mean the length of the side that”s adjacent to the angle. The same definitions, in words, can be used for angle *B* as follows:

Notice that sin *A* = cos *B* and cos *A* = sin *B*.

**SOHCAHTOA**

Here”s a word you should remember on test day so you can keep clear on the definitions of sin *θ*, cos *θ*, and tan *θ*: **SOHCAHTOA**. This isn”t some magic word to chant over your test booklet; it simply helps you remember that

** S**ine =

**pposite side over**

*O***ypotenuse**

*H*** C**osine =

**djacent side over**

*A***ypotenuse**

*H*** T**angent =

**pposite over**

*O***djacent side**

*A*The definitions

can be used for any acute angle *q* (theta) in a right triangle.

The values of the sine, cosine, and tangent of the acute angles in a 3-4-5 right triangle are listed in the specific example that follows:

sin |
sin |

cos |
cos |

tan |
tan |

We can also figure out the values of the sine, cosine, and tangent of the acute angles in a couple of special (and common) right triangles: the 30°-60° and the 45°-45° right triangles:

**Triangle Mnemonics**

A 30°-60°-90° triangle

has 3 different angles

with sides in proportions

1- -2. A 45°-45°-90°

triangle has 3 distinct

angles with sides in

proportions 1-1-.

The number of distinct

angles is what goes under

the root sign.

sin 30° = cos 60° = = 0.50

cos 30° = sin 60° = ≈ 0.87

tan 30° = ≈ 0.58, tan 60° = ≈ 1.73

sin 45° = cos 45° = ≈ 0.71

tan 45° = 1

If we know the values of these functions for other acute angles, we can use them to figure out the missing sides of a right triangle. This is one of the most common uses of trig for the physics in this book. For example, consider the triangle below with hypotenuse 5 and containing an acute angle, *θ*, of measure 30°:

Sin 30° is 0.5, so because sin*θ* = *a*/5, we can figure out that

*a* = 5 sin*θ* = 5 sin30° = 5(0.5) = 2.5

We can use the Pythagorean theorem to figure out *b*, the length of the other side. Or, if we are told that cos 30° is about 0.87, then since cos*θ*= *b*/5, we”d find that

*b* = 5 cos*θ* = 5 cos30° ≈ 5(0.87) = 4.4

This gives us

These values can be checked by the Pythagorean theorem, since

2.5^{2} + 4.4^{2} ≈ 5^{2}.

This example illustrates this important, general fact: If the hypotenuse of a right triangle is *c*, then the length of the side opposite one of the acute angles, *θ*, is *c* sin*θ*, and the length of the side adjacent to this angle is *c* cos*θ* as follows: