### All ACT Math Resources

## Example Questions

### Example Question #1 : Sine

Which of the following statements is (are) true:

I. The domain of the tangent function is all real numbers

II. The range of the sine function is all real numbers

III. The periods of the tangent, sine, and cosine functions are the same.

**Possible Answers:**

none of the above

II and III only

I and II only

II only

I only

**Correct answer:**

II only

The domain of the tangent function does not include any values of x that are odd multiples of π/2 .

The range of the sine function is from [-1, 1].

The period of the tangent function is π, whereas the period for both sine and cosine is 2π.

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

Which of the following represents a sine wave with a range of ?

**Possible Answers:**

**Correct answer:**

The range of a sine 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, then, it is fine to use . The for the function parameter only alters the period of the equation, making its waves "thinner."

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

Which of the following sine waves has a range of to ?

**Possible Answers:**

**Correct answer:**

The range of a sine 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 add to . This requires an upward 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 #4 : How To Find The Range Of The Sine

What is the range of the trigonometric function given by the equation:

**Possible Answers:**

**Correct answer:**

The range of the sine and cosine functions are the closed interval from the negative amplitude and the positive amplitude. The amplitude is given by the coefficient, in the following general equation:

. Thus we see the range is:

### Example Question #5 : How To Find The Range Of The Sine

What is the range of the following trigonometric equation:

?

**Possible Answers:**

**Correct answer:**

The range of a sine or cosine function spans from the negative amplitude to the positive amplitude. The amplitude is in the general formula:

Thus we see amplitude of our function is and so the range is:

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