### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find The Range Of The Cosine

Simplify (cosΘ – sinΘ)^{2}

**Possible Answers:**

1 + cos2Θ

1 – sin2Θ

1 + sin2Θ

cos2Θ – 1

sin2Θ – 1

**Correct answer:**

1 – sin2Θ

Multiply out the quadratic equation to get cosΘ^{2} – 2cosΘsinΘ + sinΘ^{2}

Then use the following trig identities to simplify the expression:

sin2Θ = 2sinΘcosΘ

sinΘ^{2} + cosΘ^{2} = 1

1 – sin2Θ is the correct answer for (cosΘ – sinΘ)^{2}

1 + sin2Θ is the correct answer for (cosΘ + sinΘ)^{2}

### Example Question #2 : How To Find The Range Of The Cosine

Which of the following represents a cosine function with a range of to ?

**Possible Answers:**

**Correct answer:**

The range of a cosine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at and the lowest at ; however, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift. This would make the minimum value to be and the maximum value to be .

For our question, the range of values covers . This range is accomplished by having either or as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be instead of . To do this, you will need to subtract , or , from . This requires an downward shift of . An example of performing a shift like this is:

Among the possible answers, the one that works is:

The parameter does not matter, as it only alters the frequency of the function.

### Example Question #1 : How To Find The Range Of The Cosine

Which of the following represents a cosine function with a range from to ?

**Possible Answers:**

**Correct answer:**

The range of a cosine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at and the lowest at ; however, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift. This would make the minimum value to be and the maximum value to be .

For our question, the range of values covers . This range is accomplished by having either or as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be instead of . This requires an upward shift of . An example of performing a shift like this is:

Among the possible answers, the one that works is:

### Example Question #2 : How To Find The Range Of The Cosine

What is the range of the trigonometric function defined by ?

**Possible Answers:**

**Correct answer:**

The range of a sine or cosine function spans from the negative amplitude to the positive amplitutde. The amplutide is given by in the equation . Thus the range for our function is

### Example Question #3 : How To Find The Range Of The Cosine

What is the range of the given trigonometric equation:

**Possible Answers:**

**Correct answer:**

For the sine and cosine funcitons, the range is equal to the negative amplitude to the positive amplitude.

The amplitude is found by taking from the general equation:

.

We see in our equation that

(when no coefficient is written, it is a 1).

Thus we get that the amplitude is

### Example Question #3 : How To Find The Range Of The Cosine

What is the range of the function ?

**Possible Answers:**

There is no valid range for this equation.

**Correct answer:**

The range of the function represents the spread of possible answers you can get for , given all values of . In this case, the ordinary range for a cosine function is , since the largest value that cosine can solve to is (for a cosine of or a multiple of one of those values), and the smallest value cosine can solve to is (for a cosine of or a multiple of one of those values).

However, in this case our final answer is increased by after the cosine is applied to . This results in a final range of to (or, in other words, to , plus ).

So, our final range is .

### Example Question #52 : Cosine

What is the range of the function ?

**Possible Answers:**

There is no range that fits this equation.

**Correct answer:**

The range of the function represents the spread of possible answers you can get for , given all values of . In this case, the ordinary range for a cosine function is , since the largest value that cosine can solve to is (for a cosine of or a multiple of one of those values), and the smallest value cosine can solve to is (for a cosine of or a multiple of one of those values).

However, in this case our final answer is multiplied by -3 after the cosine is applied to . This results in a final range of to (or, in other words, to , multiplied by ).

So, our final range is .

### Example Question #8 : How To Find The Range Of The Cosine

What is the range of the function ?

**Possible Answers:**

There is no range that fits this function.

**Correct answer:**

The range of the function represents the spread of possible answers you can get for , given all values of . In this case, the ordinary range for a cosine function is , since the largest value that cosine can solve to is (for a cosine of or a multiple of one of those values), and the smallest value cosine can solve to is (for a cosine of or a multiple of one of those values).

However, in this case our final answer is first multiplied by , then decreased by after the cosine is applied to . Multiplying the initial range by results in a new range of . Next, subtracting from this range gives us a new range of .

**Note that the does not change our range. **This is because, irrespective of other multipliers, a cosine operation can only return values between and . To think of this a different way, will give us the same returns as , only we will move around the unit circle five times as much before finding our answer.

Thus, our final range is .

### Example Question #4 : How To Find The Range Of The Cosine

Which of the following functions has a range of ?

**Possible Answers:**

None of these have the specified range.

**Correct answer:**

One fast way to match a range to a function is to look for the function which has a vertical shift equal to the mean of the range values. In other words, for the standard trigonometric function , where represents the vertical shift, .

In this case, since our range is , we expect our to be .

Of the answer choices, only has , so we know this is our correct choice.

### Example Question #10 : How To Find The Range Of The Cosine

Which of the following functions has a range of ?

**Possible Answers:**

None of these formulas has the specified range.

**Correct answer:**

One fast way to match a range to a function is to look for the function which has a vertical shift equal to the mean of the range values. In other words, for the standard trigonometric function , where represents the vertical shift, .

In this case, since our range is , we expect our to be .

Of the answer choices, only has , so we know this is our correct choice.