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.
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 × 108. 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 × 10n, 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 10m × 10n = 10m + n and 10m/10n = 10m – n. So, for example, (3 × 108)(2.5 × 10–12) = 7.5 × 10–4 and (8 × 109)/(2 × 10–5) = 4 × 1014.
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):
a2 + b2 = c 2
Triangles and Beans?
actually a cult leader
around 500 B.C. Some rules
of his number-worshipping
group included prohibitions
against eating beans and
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.
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
Sine = Opposite side over Hypotenuse
Cosine = Adjacent side over Hypotenuse
Tangent = Opposite over Adjacent side
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 A =
sin B =
cos A =
cos B =
tan A =
tan B =
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:
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
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 sinθ = a/5, we can figure out that
a = 5 sinθ = 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 cosθ= b/5, we”d find that
b = 5 cosθ = 5 cos30° ≈ 5(0.87) = 4.4
This gives us
These values can be checked by the Pythagorean theorem, since
2.52 + 4.42 ≈ 52.
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 sinθ, and the length of the side adjacent to this angle is c cosθ as follows: