SAT Math 1 & 2 Subject Tests

Chapter 8

Trigonometry

The rules of trigonometry tested on the Math Level 1 Subject Test are much more limited than those tested on the Math Level 2 Subject Test. Trigonometry on the Math Level 1 Subject Test is confined to right triangles and the most basic relationships between the sine, cosine, and tangent functions. If you’re taking the Math Level 1, that’s the only material from this chapter you need to know. If you plan to take the Math Level 2, then this entire chapter is your domain; rule it wisely.

DEFINITIONS

Here are some trigonometric terms that appear on the Math Subject Tests. Make sure you’re familiar with them. If the meaning of any of these vocabulary words keeps slipping your mind, add that word to your flash cards.

Acute Angle

An angle whose measure in degrees is between 0 and 90, exclusive.

Obtuse Angle

An angle whose measure in degrees is between 90 and 180, exclusive.

Theta

The symbol θ (pronounced thay-tuh) is a variable, just like x and y, used to represent the measure of an angle in trigonometry.

arc−

Prefix added to trigonometric functions, meaning inverse.

THE BASIC FUNCTIONS

The basis of trigonometry is the relationship between the parts of a right triangle. When you know the measure of one of the acute angles in a right triangle, you know all the angles in that triangle. For example, if you know that a right triangle contains a 20° angle, then you know all three angles—the right triangle must have a 90° angle, and because there are 180° in a triangle, the third angle must measure 70°. You don’t know the lengths in the triangle, but you know its shape and its proportions.

Similar Right Triangles

Remember that similar
triangles have the same
angles. So, any right
triangle that contains a 20°
angle will be similar to all
other right triangles with a
20° angle.

A right triangle that contains a 20° angle can have only one shape, though it can be any size. The same is true for a right triangle containing any other acute angle. That’s the fundamental idea of trigonometry. Once you know the measure of an acute angle in a right triangle, you know that triangle’s proportions.

The three basic functions in trigonometry—the sine, cosine, and tangent—are ways of expressing proportions in a right triangle (that’s the ratio of one side to another). They may sound familiar to you. Or maybe you’ve heard of a little phrase called SOHCAHTOA?

Let’s break it down.


SOHCAHTOA


Sine

The sine of an angle is the ratio of the opposite side to the hypotenuse. The sine function of an angle θ is abbreviated sin θ.

It’s All About
Proportions

A trigonometric function
of any angle comes from
the proportions of a right
triangle containing that
angle. For any given angle,
there is only one possible
set of proportions.

Cosine

The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The cosine function of an angle θ is abbreviated cos θ.

Tangent

The tangent of an angle is the ratio of the opposite side to the adjacent side. The tangent function of an angle θ is abbreviated tan θ.

These three functions form the basis of everything else in trigonometry. All of the more complicated functions and rules in trigonometry can be derived from the information contained in SOHCAHTOA.

What Your Calculator Can Do for You

Tables of sine, cosine, and tangent values are programmed into your calculator—that’s what the “sin,” “cos,” and “tan” keys do.

•  If you press one of the three trigonometric function keys and then enter an angle measure, your calculator will give you the function (sine, cosine, or tangent) of that angle. Just make sure that your calculator is in degree mode. This operation is written:
sin 30° = 0.5     cos 30° = 0.866     tan 30° = 0.577

•  Your calculator can also take a trig function value and tell you what angle would produce that value. Press the “2nd” key, then press “sin,” “cos,” or “tan,” then enter the decimal or fraction you’re given, and your calculator will give you the measure of that angle. This is called taking an inverse function, and it’s written:

The expressions “sin−1 (0.5)” and “arcsin (0.5)” have the same meaning. Both mean “the angle whose sine is 0.5.” While ordinary trig functions take angle measures and output ratios, inverse trig functions take ratios and produce the corresponding angle measures; they work in reverse.

Check Your Calculator

For some scientific calculators,
you need to enter
things in reverse order. To
find sin 30°, for example,
you would type “30” first
and then hit “sin.” To find
sin−1(0.5), you would type
“0.5” first and then hit
“2nd” and “sin.”

Finding Trig Functions in Right Triangles

On the Math Level 1 Subject Test, the three basic trigonometric functions always occur in right triangles—particularly the Pythagorean triplets from Chapter 5.

Special Right Triangles

Be on the lookout for
special right triangles on
trigonometry questions!

DRILL

Use the definitions of the sine, cosine, and tangent to fill in the requested quantities in the following triangles. The answers to these drills can be found in Chapter 12.

  1.  sin θ = _____________

cos θ = _____________

tan θ = _____________

  2.  sin θ = _____________

cos θ = _____________

tan θ = _____________

  3.  sin θ = _____________

cos θ = _____________

tan θ = _____________

  4.  sin θ = _____________

cos θ = _____________

tan θ = _____________

Completing Triangles

The preceding examples have all involved figuring out the values of trigonometric functions from lengths in a right triangle. Slightly more difficult trigonometry questions may require you to go the other way and figure out lengths or measures of angles using trigonometry. For example:

x = _____________

Check Your Mode

For Math Level 1, your
calculator should always
be in degree mode. For
Math Level 2, it may
sometimes need to be in
radian mode (more on that
later).

Because we’re dealing with the hypotenuse and the side that is opposite the angle, the best definition to use is sine.

sin = 

sin 35° =                    

5(sin 35°) = x                        

5(0.5736) = x                        

2.8679 = x                    

BC of ∆ABC therefore has a length of 2.87.


The Unknown

In triangle ABC, you know only two quantities—the length of AB and the measure of ∠A. This question, unlike previous examples, doesn’t give you enough information to use the Pythagorean theorem. What you need is an equation that relates the information you have (AB and ∠A) to the information you don’t have (x). Use the SOHCAHTOA definitions to set up an equation. Solve that equation, and you find the value of the unknown.


You can use a similar technique to find the measure of an unknown angle in a right triangle. For example:

x = ________________________

In triangle DEF, you know DF and DFEF is the side that is opposite the angle we’re looking for, and DF is the side that is adjacent to that same angle. So the best definition to use is tangent.

tan = 

tan x =      

tan x =          

tan x = 0.5      

To solve for x, take the inverse tangent of both sides of the equation. On the left side, that just gives you x. The result is the angle whose tangent is 0.5.

tan−1 (tan x) = tan−1(0.5)

x = 26.57°

Let Your Calculator Help

To take the inverse tangent
of the right side of this
equation, press the “2nd”
key, press the “tan” key,
and then type in 0.5.

The measure of ∠D is therefore 26.57°.

DRILL

Use the techniques you’ve just reviewed to complete the following triangles. The answers to these drills can be found in Chapter 12.

  1.  AB = _____________

CA = _____________

B = _____________

  2.  EF = _____________

FD = _____________

D = _____________

  3.  HJ = _____________

JK = _____________

J = _____________

  4.  LM = _____________

MN = _____________

N = _____________

  5.  TR = _____________

S = _____________

T = _____________

  6.  YW = _____________

W = _____________

Y = _____________

TRIGONOMETRIC IDENTITIES

Some Math Subject Test questions will ask you to do algebra with trigonometric functions. These questions usually involve using the SOHCAHTOA definitions of sine, cosine, and tangent. Often, the way to simplify equations that are mostly made up of trigonometric functions is to express the functions as follows:



Writing trig functions this way can simplify trig equations, as the following example shows:

Working with trig functions this way lets you simplify expressions. The equation above is actually a commonly used trigonometric identity. You should memorize this, as it can often be used to simplify equations.


 = tan x


Here’s the breakdown of another frequently used trigonometric identity:

That last step may seem a little baffling, but it’s really simple. This equation is based on a right triangle, in which O and A are legs of the triangle, and H is the hypotenuse. Consequently you know that O2 + A2 = H2. That’s just the Pythagorean theorem. That’s what lets you do the last step, in which  = 1. This completes the second commonly used identity that you should memorize.


sin2 θ + cos 2 θ = 1


In addition to memorizing these two identities, you should practice working algebraically with trig functions in general. Some questions may require you to use the SOHCAHTOA definitions of the trig functions; others may require you to use the two identities you’ve just reviewed. Take a look at these examples:

35.  If sin x = 0.707, then what is the value of (sin x) • (cos x) • (tan x) ?

(A)   1.0

(B)   0.707

(C)   0.5

(D)   0.4

(E)   0.207

Here’s How to Crack It

This is a tricky question. To solve it, simplify that complicated trigonometric expression. Writing in the SOHCAHTOA definitions works just fine, but in this case it’s even faster to use one of those identities.

Now it’s a simpler matter to answer the question. If sin x = 0.707, then sin2 x = 0.5. The answer is (C).

Take a look at this one:

36.  If sin a = 0.4, and 1 − cos2 a = x, then what is the value of x ?

(A)   0.8

(B)   0.6

(C)   0.44

(D)   0.24

(E)   0.16

Here’s How to Crack It

Here again, the trick to the question is simplifying the complicated trig expression. Since sin2 θ + cos2 θ= 1, you can rearrange any of those terms to rephrase it. Using the second trig identity, you can quickly take these steps:

1 − cos2 a = x

sin2 a = x      

(0.4)2 = x       

x = 0.16       

And that’s the answer. (E) is correct.

Using the SOHCAHTOA definitions and the two trigonometric identities reviewed in this section, simplify trigonometric expressions to answer the following sample questions.

DRILL

Try the following practice questions. The answers to these drills can be found in Chapter 12.

25.  (1 − sin x)(1 + sin x) =

(A)   cos x

(B)   sin x

(C)   tan x

(D)   cos2 x

(E)   sin2 x

31.   =

(A)   

(B)   

(C)   1

(D)   cos2 x

(E)   tan x

39.   −(sin x)(tan x) =

(A)   cos x

(B)   sin x

(C)   tan x

(D)   cos2 x

(E)   sin2 x

42.   =

(A)   1 − cos x

(B)   1 − sin x

(C)   tan x + 1

(D)   cos2 x

(E)   sin2 x

The Other Trig Functions

On the Math Level 2 Subject Test, you may run into the other three trigonometric functions—the cosecant, secant, and cotangent. These functions are abbreviated cscθ, secθ, and cotθ, respectively, and they are simply the reciprocals of the three basic trigonometric functions you’ve already reviewed.

Here’s how they relate:



You can also express these functions in terms of the sides of a right triangle—just by flipping over the SOHCAHTOA definitions of the three basic functions.



These three functions generally show up in algebra-style questions, which require you to simplify complex expressions containing trig functions. The goal is usually to get an expression into the simplest form possible, one that contains no fractions. Such questions are like algebra-style questions involving the three basic trig functions; the only difference is that the addition of three more functions increases the number of possible forms an expression can take. For example:

The entire expression (cos x)(cot x) + (sin2 x csc x) is therefore equivalent to a single trig function, the cosecant of x. That’s generally the way algebraic trigonometry questions work on the Math Level 2 Subject Test.

Drill

Simplify each of these expressions to a single trigonometric function. Keep an eye out for the trigonometric identities reviewed on this page; they’ll still come in handy. The answers to these drills can be found in Chapter 12.

19.  sec2 x − 1 =

(A)   sin x cos x

(B)   sec2 x

(C)   cos2 x

(D)   sin2 x

(E)   tan2 x

23.   =

(A)   cos x

(B)   sin x

(C)   tan x

(D)   sec x

(E)   csc x

24.  sin x + (cos x)(cot x) =

(A)   csc x

(B)   sec x

(C)   cot x

(D)   tan x

(E)   sin x

GRAPHING TRIGONOMETRIC FUNCTIONS

There are two common ways to represent trigonometric functions graphically—on the unit circle, or on the coordinate plane (you’ll get a good look at both methods in the coming pages). Both of these graphing approaches are ways of showing the repetitive nature of trigonometric functions. All of the trig functions (sine, cosine, and the rest) are called periodic functions. That simply means that they cycle repeatedly through the same values.

The Unit Circle


What Goes Around Comes Around

If you picked a certain angle and its sine, cosine, and tangent, and then slowly changed the measure of that angle, you’d see the sine, cosine, and tangent change as well. But after a while, you would have increased the angle by 360°—in other words, you would come full circle, back to the angle you started with, going counterclockwise. The new angle, equivalent to the old one, would have the same sine, cosine, and tangent as the original. As you continued to increase the angle’s measure, the sine, cosine, and tangent would cycle through the same values all over again. All trigonometric functions repeat themselves every 360°. The tangent and cotangent functions actually repeat every 180°.

Thus, angles of 0° and 360° are mathematically equivalent. So are angles of 40° and 400°, or 360° and 720°. Any two angle measures separated by 360° are equivalent. For example, to find equivalent angles to 40°, you just keep adding 360°. Likewise, you can go around the unit circleclockwise by subtracting multiples of 360°. Some angles equivalent to 40° would thus be 40° − 360° = −320°, −680°, −1040°, and so on. In the next few sections, you’ll see how that’s reflected in the graphs of trigonometric functions.


This is the unit circle. It looks a little like the coordinate plane; in fact, it is the coordinate plane, or at least a piece of it. The circle is called the unit circle because it has a radius of 1 (a single unit). This is convenient because it makes trigonometric values easy to figure out. The radius touching any point on the unit circle is the hypotenuse of a right triangle. The length of the horizontal leg of the triangle is the cosine (which is therefore the x-coordinate) and the length of the vertical leg is the sine (which is the y-coordinate). It works out this way because sine = opposite ÷ hypotenuse, and cosine = adjacent ÷ hypotenuse; and here the hypotenuse is 1, so the sine is simply the length of the opposite side, and the cosine simply the length of the adjacent side.

Suppose you wanted to show the sine and cosine of a 30° angle. That angle would appear on the unit circle as a radius drawn at a 30° angle to the positive x-axis (above). The x-coordinate of the point where the radius intercepts the circle is 0.866, which is the value of cos 30°. The y-coordinate of that point is 0.5, which is the value of sin 30°.

Now take a look at the sine and cosine of a 150° angle. As you can see, it looks just like the 30° angle, flipped over the y-axis. Its y-value is the same—sin 150° = 0.5—but its x-value is now negative. The cosine of 150° is −0.866.

Here, you see the sine and cosine of a 210° angle. Once again, this looks just like the 30° angle, but this time flipped over the x- and y-axes. The sine of 210° is −0.5; the cosine of 210° is −0.866.

This is the sine and cosine of a 330° angle. Like the previous angles, the 330° angle has a sine and cosine equivalent in magnitude to those of the 30° angle. In the case of the 330° angle, the sine is negative and the cosine positive. So, sin 330° = −0.5 and cos 330° = 0.866. Notice that a 330° angle is equivalent to an angle of −30°.

Following these angles around the unit circle gives us some useful information about the sine and cosine functions.


•  Sine is positive between 0° and 180° and negative between 180° and 360°. At 0°, 180°, and 360°, sine is zero. At 90°, sine is 1. At 270°, sine is −1.

•  Cosine is positive between 0° and 90° and between 270° and 360°. (You could also say that cosine is positive between −90° and 90°.) Cosine is negative between 90° and 270°. At 90° and 270°, cosine is zero. At 0° and 360°, cosine is 1. At 180°, cosine is −1.


When these angles are sketched on the unit circle, sine is positive in quadrants I and II, and cosine is positive in quadrants I and IV. There’s another important piece of information you can get from the unit circle. The biggest value that can be produced by a sine or cosine function is 1. The smallest value that can be produced by a sine or cosine function is −1.

Following the tangent function around the unit circle also yields useful information.

The sine of 45° is , or 0.707, and the cosine of 45° is also , or 0.707. Since the tangent is the ratio of the sine to the cosine, that means that the tangent of 45° is 1.

The tangent of 135° is −1. Here the sine is positive, but the cosine is negative.

The tangent of 225° is 1. Here the sine and cosine are both negative.

The tangent of 315° is −1. Here the sine is negative, and the cosine is positive.

This is the pattern that the tangent function always follows. It’s positive in quadrants I and III and negative in quadrants II and IV.


•  Tangent is positive between 0° and 90° and between 180° and 270°.

•  Tangent is negative between 90° and 180° and between 270° and 360°.


The unit circle is extremely useful for identifying equivalent angles (like 270° and −90°), and also for seeing other correspondences between angles, like the similarity between the 45° angle and the 135° angle, which are mirror images of one another on the unit circle.

A good way to remember where sine, cosine, and tangent are positive is to write the words of the phrase All Students Take Calculus in quadrants I, II, III, and IV, respectively, on the coordinate plane. The first letter of each word (A S T C) tells you which functions are positive in that quadrant. So All three functions are positive in quadrant I, the Sine function is positive in quadrant II, the Tangent function is positive in quadrant III, and the Cosine function is positive in quadrant IV.

Drill

Make simple sketches of the unit circle to answer the following questions about angle equivalencies. The answers to these drills can be found in Chapter 12.

18.  If sin 135° = sin x, then x could equal

(A)   −225°

(B)   −45°

(C)   225°

(D)   315°

(E)   360°

21.  If cos 60° = cos n, then n could be

(A)   30°

(B)   120°

(C)   240°

(D)   300°

(E)   360°

26.  If sin 30° = cos t, then t could be

(A)   −30°

(B)   60°

(C)   90°

(D)   120°

(E)   240°

30.  If tan 45° = tan x, then which of the following could be x ?

(A)   −45°

(B)   135°

(C)   225°

(D)   315°

(E)   360°

36.  If 0° ≤ θ ≤ 360° and (sin θ)(cos θ) < 0, which of the following gives the possible values of θ ?

(A)   0° ≤ θ ≤ 180°

(B)   0° ≤ θ ≤ 180° or 270° ≤ θ ≤ 360°

(C)   0° < θ < 90° or 180° < θ < 270°

(D)   90° < θ < 180° or 270° < θ < 360°

(E)   0° < θ < 180° or 270° < θ < 360°

Degrees and Radians

On the Math Level 2 Subject Test, you may run into an alternate means of measuring angles. This alternate system measures angles in radians rather than degrees. One degree is defined as  of a full circle. One radian, on the other hand, is the measure of an angle that intercepts an arc exactly as long as the circle’s radius. Since the circumference of a circle is 2π times the radius, the circumference is about 6.28 times as long as the radius, and there are about 6.28 radians in a full circle.

Because a number like 6.28 isn’t easy to work with, angle measurements in radians are usually given in multiples or fractions of π. For example, there are exactly 2π radians in a full circle. There are π radians in a semicircle. There are  radians in a right angle. Because 2π radians and 360° both describe a full circle, you can relate degrees and radians with the following proportion:



To convert degrees to radians, just plug the number of degrees into the proportion and solve for radians. The same technique works in reverse for converting radians to degrees. The figures on the next page show what the unit circle looks like in radians, compared to the unit circle in degrees.

DRILL

By referring to these unit circles and using the proportion given on this page, fill in the following chart of radian−degree equivalencies. The answers to these drills can be found in Chapter 12.

The Shift

A scientific or graphing
calculator can calculate
trigonometric functions of
angles entered in radians,
as well. However, it is
necessary to shift the calculator
from degree mode
into radian mode. Consult
your calculator’s operating
manual and make sure you
know how to do this.

Degrees

Radians

30°

 

45°

 
 

 

120°

 
 

150°

 
 

π

 

240°

 
 

300°

 

315°

330°

 
 

Trigonometric Graphs on the Coordinate Plane

In a unit-circle diagram, the x-axis and y-axis represent the horizontal and vertical components of an angle, just as they do on the coordinate plane. The angle itself is represented by the angle between a certain radius and the positive x-axis. Any trigonometric function can be represented on a unit-circle diagram.

Periodic Repetitions

Trigonometric functions
are called periodic functions.
The period of a
function is the distance a
function travels before it
repeats. A periodic function
will repeat the same
pattern of values forever.
As you can see from the
graph, the period of the
sine function is 2π radians,
or 360°.

When a single trigonometric function is graphed, however, the axes take on different meanings. The x-axis represents the value of the angle; this axis is usually marked in radians. The y-axis represents a specific trigonometric function of that angle. For example, here is the coordinate plane graph of the sine function.

Compare this graph to the unit circle on this page. A quick comparison will show you that both graphs present the same information. At an angle of zero, the sine is zero; at a quarter circle ( radians, or 90°), the sine is 1; and so on.

Make Things Easier

Because the sine and
cosine curves have the
same shape and size, you
can focus on memorizing
the facts for just one
of them.

Here is the graph of the cosine function.

Notice that the cosine curve is identical to the sine curve, only shifted to the left by  radians, or 90°. The cosine function also has a period of 2π radians.

Finally, here is the graph of the tangent function.

This function, obviously, is very different from the others. First, the tangent function has no upper or lower limit, unlike the sine and cosine functions, which produce values no higher than 1 or lower than −1. Second, the tangent function has asymptotes. These are values on the x-axis at which the tangent function does not exist; they are represented by vertical dotted lines. Finally, the tangent function has a period of π radians.


The Undefined Tangent

It’s easy to see why the tangent function’s graph has asymptotes, if you recall the definition of the tangent.

tan θ = 

A fraction is undefined whenever its denominator equals zero. At any value where the cosine function equals zero, therefore, the tangent function is undefined—it doesn’t exist. As you can see by comparing the cosine and tangent graphs, the tangent has an asymptote wherever the cosine function equals zero.


It’s important to be able to recognize the graphs of the three basic trigonometric functions. You’ll find more information about these functions and their graphs in the following chapter on functions.

TRIGONOMETRY IN NON-RIGHT TRIANGLES

The rules of trigonometry are based on the right triangle, as you’ve seen in the preceding sections. Right triangles are not, however, the only places you can use trigonometric functions. There are a couple of powerful rules relating angles and lengths that you can use in any triangle. These are rules that only come up on the Math Level 2 Subject Test, and there are only two basic laws you need to know—the Law of Sines and the Law of Cosines.

The Law of Sines

The Law of Sines can be used to complete the dimensions of a triangle about which you have partial information. This is what the law says:

Please Explain

In English, this law means
that the sine of each angle
in a triangle is related to
the length of the opposite
side by a constant proportion.
Once you figure out
the proportion relating the
sine of one angle to the
opposite side, you know
the proportion for
every angle.



Let’s take a look at an example.

B = _____________

AB = _____________

AC = _____________

In this triangle, you know only two angles and one side. Immediately, you can fill in the third angle, knowing that there are 180° in a triangle. Then, you can fill in the missing sides using the Law of Sines. Write out the proportions of the Law of Sines, filling in the values you know.

We Know, We Know

Yes, 0.643 ÷ 8 rounds to
0.0804, but if you keep
the value of sin 40° in your
calculator and divide by 8,
you’ll get 0.0803.

At this point, you can set up two individual proportions and solve them individually for b and c, respectively.

The length of AB is therefore 6.23, and the length of AC is 11.70. Now you know every dimension of triangle ABC.


The Law of Sines can be used in any triangle if you know

•  two sides and one of their opposite angles (this gives you two different possible triangles)

•  two angles and any side


The Law of Cosines

When you don’t have the information necessary to use the Law of Sines, you may be able to use the Law of Cosines instead. The Law of Cosines is another way of using trigonometric functions to complete partial information about a triangle’s dimensions.


c2 = a2 + b2 −2ab cos C


The Law of Cosines is a way of completing the dimensions of any triangle. You’ll notice that it looks a bit like the Pythagorean theorem. That’s basically what it is, with a term added to the end to compensate for non-right angles. If you use the Law of Cosines on a right triangle, the “2ab cosC” term becomes zero, and the law becomes the Pythagorean theorem. The Law of Cosines can be used to fill in unknown dimensions of a triangle when you know any three of the quantities in the formula.

c = ______________ A = ______________ B = ______________

In this triangle, you know only two sides and an angle—the angle between the known sides. That is, you know ab, and C. In order to find the length of the third side, c, just fill the values you know into the Law of Cosines, and solve.

c2 = a2 + b2 − 2ab cos C

c2 = (10)2 + (12)2 − 2(10)(12) cos 45°

c2 = 100 + 144 − 240(0.707)

c2 = 74.3

c2 = 8.62

The length of AB is therefore 8.62. Now that you know the lengths of all three sides, just use the Law of Sines to find the values of the unknown angles, or re-arrange the Law of Cosines to put the other unknown angles in the C position, and solve to find the measures of the unknown angles.


The Law of Cosines can be used in any triangle if you know

•  all three sides

•  two sides and the angle between them


DRILL

In the following practice exercises, use the Law of Sines and the Law of Cosines to complete the dimensions of these non-right triangles. The answers to these drills can be found in Chapter 12.

  1.  a = ______________ B = ______________ C = ______________

  2.  A = ______________ B = ______________ C = ______________

  3.  c = ______________ B = ______________ C = ______________

POLAR COORDINATES

Polar coordinates are another way of describing the position of a point in the coordinate plane. In the previous figure, the position of point P can be described in two ways. In standard rectangular coordinates, you would count across from the origin to get an x-coordinate and up from the origin to get a y-coordinate. (Remember: These x and y distances can be regarded as legs of a right triangle. The hypotenuse of the triangle is the distance between the point and the origin.) Rectangular coordinates consist of a horizontal distance and a vertical distance, and take the form (x,y). In rectangular coordinates, point P would be described as (5, 5).

Polar coordinates consist of the distance, r, between a point and the origin, and the angle, θ, between that segment and the positive x-axis. Polar coordinates thus take the form (rθ). The angle θ can be expressed in degrees, but is more often expressed in radians. In polar coordinates, therefore, P could be described as (10, 30°) or .

As you saw in the unit circle, there’s more than one way to express any angle. For any angle, there is an infinite number of equivalent angles that can be produced by adding or subtracting 360° (or 2π, if you’re working in radians) any number of times. Therefore, there is an infinite number of equivalent polar coordinates for any point. Point P, at (10, 30°), can also be expressed as (10, 390°), or . You can continually produce equivalent expressions by adding or subtracting 360° (or 2π).

There’s still another way to produce equivalent polar coordinates. The distance from the origin—the r in (rθ)—can be negative. This means that once you’ve found the angle at which the hypotenuse must extend, a negative distance extends in the opposite direction, 180° away from the angle. Therefore, you can also create equivalent coordinates by increasing or decreasing the angle by 180° and flipping the sign on the distance. The point P(10, 30°) or  could also be expressed as (−10, 210°) or  Other equivalent coordinates can be generated by pairing equivalent angles with these negative distances.

Converting Coordinates

Converting rectangular coordinates to polar coordinates and vice versa is simple. You just use the trigonometry techniques reviewed in this chapter.

Given a point (rθ) in polar form, you can find its rectangular coordinates by drawing a right triangle such as the following:

From this picture, using SOHCAHTOA and the Pythagorean theorem, you can see the following relationships:


cos θ = ;  sin θ = ;  tan θ = ;  x2 + y2 = r2;  θ = tan−1 


DRILL

Try the following practice questions about polar coordinates. The answers to these drills can be found in Chapter 12.

39.  Which of the following rectangular coordinate pairs is equivalent to the polar coordinates ?

(A)   (0.5, 1.7)

(B)   (2.6, 5.2)

(C)   (3.0, 5.2)

(D)   (4.2, 4.8)

(E)   (5.2, 15.6)

42.  The point  in polar coordinates is how far from the x-axis?

(A)   3.67

(B)   4.95

(C)   5.25

(D)   6.68

(E)   16.71

45.  The points AB, and C in polar coordinates define which of the following?

(A)   A point

(B)   A line

(C)   A plane

(D)   A three-dimensional space

(E)   None of these

Summary

·        For the purposes of the Level 1 Subject Test, trigonometry questions will deal only with basic trig.

·        Memorize SOHCAHTOA. It’s your best friend. sin = cos = , and tan = . Tan is also equal to . ETS will test these with trigonometric identity questions.

·        You can use the inverse of a function on your calculator to find the angle when you know the value of the corresponding trigonometric function.

·        The unit circle is a circle on the coordinate plane with a radius of 1. You can use Pythagorean theorem, SOHCAHTOA, and the fact that if you draw a line from the origin to any point on the circle and create a triangle, the hypotenuse will always be 1.

·        For the Level 2 Subject Test only, it is important to know the following:

·        The reciprocals of the trig functions are cosecant, secant, and cotangent. Their relation to the trig functions are: csc θ = , sec θ = , and cot θ = .

·        Radians are just another way to measure angles. The relationship between degrees and radians is: .

·        Use All Students Take Calculus to remember which trig functions are positive in each quadrant.

·        The graphs of trigonometric functions are periodic functions. Know what each graph looks like.

·        For non-right triangles, there are two important laws. In a triangle with sides ab, and c and corresponding angles AB, and C, the Law of Sines says that ; and the Law of Cosines says that c2 = a2 + b2 − 2ab cos C.

·        Polar coordinates use the distance, r, between a point and the origin, and the angle θ, which can be written in degrees or radians. A point in polar coordinates would be (rθ).

·        When converting coordinates between rectangular and polar, use either SOHCAHTOA and the Pythagorean theorem, or the following relationships: cos θ = ; sin θ = ; tan θ = x2 + y2 = r2; and θ tan−1.