Cracking the SAT subject test in math 2, 2nd edition

Cracking the SAT Subject Test in Math 2, 2nd Edition (B) sinx (C) tanx (D) secx (E) cscx 24 sinx + (cos x)(cotx) = (A) cscx (B) secx (C) cotx (D) tanx (E) sinx 38 = (A) 1 (B) csc2x (C) sinx cosx (D) t[.]

(B) sinx (C) tanx (D) secx (E) cscx

24 sinx + (cos x)(cotx) =

(A) cscx (B) secx (C) cotx (D) tanx (E) sinx

(A) 1 (B) csc2x

(C) sinx cosx

(D) tan2x

(E) cot2x

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

Do Not Adjust Your Math Book

Some of the images of the unit circle on the next several pages are not drawn to scale This

reflects something you’ll find on the SAT

Subject Test itself: given a choice between the math and the illustration, always trust the math.

The Unit Circle

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 is 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 sin30°

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—sin150° = 0.5—but its x-value is now negative The cosine of 150°

is −0.866

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 circle clockwise 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

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

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, sin330° = −0.5 and cos330° = 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 SoAll three functions are positive in quadrant I, the Sine function is positive in quadrant II, the Tangent function is positive in quadrant III, and theCosine function is positive in quadrant IV.

DRILL 5: ANGLE EQUIVALENCIES

Make simple sketches of the unit circle to answer the following questions about angle equivalencies The answers can be found in Part IV

18 If sin135° = sinx, then x could equal

(A) −225°

(B) −45°

(C) 225°

(D) 315°

(E) 360°

21 If cos60° = cosn, then n could be

(A) 30°

(B) 120°

(C) 240°

(D) 300°

(E) 360°

26 If sin30° = cost, then t could be

(A) −30°

(B) 60°

(C) 90°

(D) 120°

(E) 240°

30 If tan45° = tanx, then which of the following could be x ?

(A) −45°

(B) 135°

(C) 225°

(D) 315°

(E) 360°

36 If 0° ≤ θ ≤ 360° and (sinθ)(cosθ) < 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°

40 Which of the following are equivalent to cot40°?

I cot220°

II cot130°

III

(A) I only (B) II only (C) III only (D) I and II only (E) I and III only

Degrees and Radians

On the SAT Subject Test in Math 2, 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 below show what the unit circle looks like in radians, compared to the unit circle in degrees

Radians

DRILL 6: DEGREES AND RADIANS

Fill in the following chart of radian–degree equivalencies The answers can be found in Part IV

Degrees Radians

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

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.

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°.

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

Here is the graph of the cosine function.

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

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