Trigonometry Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 8
1. Of the four quantities shown here, the last one, 367 x 103, appears in a format that fails to adhere to commonly accepted mathematical or scientific notation. Although the quantity makes sense, we should render it as 3.67 x 105 if we want to express it properly. The correct choice is D. The other three choices are okay. Choice A is in standard power-of-10 notation. Choice B is a "plain old decimal" expression. Choice C appears in alternative power-of-10 notation.
2. The uppercase E in this expression translates literally into the words "times 10 to the power of." Therefore

3.5678E+2 = 3.5678 x 102
= 356.78

The answer is A.

3. When we want to reduce a quantity by a single order of magnitude, we divide it by 10. To make a quantity four orders of magnitude smaller, we divide it by 10 four times; that is, we divide it by 104. In this case, we want to reduce 2.558 x 106 by four orders of magnitude. We have

2.558 x 106 / 104
= 2.558 x 10(6-4)
= 2.558 x 102
= 255.8

The correct choice is B.

4. When we want to multiply two quantities in scientific notation, we should carry out the following steps in order:
  • Multiply the coefficients by each other
  • Multiply the powers of 10 by each other
  • Combine the resulting coefficient and power of 10
  • Simplify the expression to standard power-of-10 notational form if necessary
  • Round off the coefficient to the correct number of significant figures if necessary

In this case, we start with the product

(4.673 x 106) x (8.77 x 10-11)

When we multiply the coefficients, we get

4.673 x 8.77 = 40.98221

When we multiply the powers of 10, we get

106 x 10-11 = 10[6+(-11)]
= 10(6-11)
= 10-5

When we combine the product of the coefficients and the product of the powers of 10, we come up with

40.98221 x 10-5

To put this expression in standard power-of-10 form, we must move the decimal point one place to the left in the coefficient; that's the equivalent of dividing by 10. To compensate, we must multiply the power-of-10 part of the expression by 10; that means we must increase the exponent by 1. The result is

4.098221 x 10-4

We're only entitled to three significant figures in the final expression, because that's the smaller number of significant figures in the two factors. When we round off the above expression to three significant figures, we get

4.10 x 10-4

The answer is C.

5. When we want to divide one quantity by another in scientific notation, we should carry out the following steps in order:
  • Divide the first coefficient by the second one
  • Divide the first power of 10 by the second one
  • Combine the resulting coefficient and power of 10
  • Simplify the expression to standard power-of-10 notational form if necessary
  • Round off the coefficient to the correct number of significant figures if necessary

We begin with the quotient

(4.673 x 106) / (8.77 x 10-11)

When we divide the first coefficient by the second one, we get approximately

4.673 / 8.77 = 0.5328392

(Here, we go out to seven significant figures; that's an arbitrary choice, and we'll round it off at the end of the process.) When we divide the first power of 10 by the second one, we get

106 / 10-11 = 10[6-(-11)]
= 10(6+11)
= 1017

When we combine the product of the coefficients and the product of the powers of 10, we come up with

0.5328392 x 1017

To put this expression in standard power-of-10 form, we must move the decimal point one place to the right in the coefficient; that's the equivalent of multiplying by 10. To compensate, we must divide the power-of-10 part of the expression by 10; that means we must decrease the exponent by 1. When we do those things, we obtain

5.328392 x 1016

We're only entitled to three significant figures in the final expression, because that's the smaller number of significant figures in the two original expressions. When we round off the above quantity to three significant figures, we get

5.33 x 1016

The correct choice is D.

6. We want to find the sum of the same two quantities as we worked with in the previous two problems. That sum is

(4.673 x 106) + (8.77 x 10-11)

The first quantity exceeds the second quantity by more than 16 orders of magnitude (a factor of 5.33 x 1016, in fact, as we discovered when we solved the previous problem). That's far more than the number of significant digits in either figure. When we encounter a situation of that sort, we can consider the sum of the two quantities as equal to the larger one, including all of its significant figures. In this case, it's 4.673 x 106. The answer is A.

7. When I use my personal computer's calculator in the scientific mode and set it for degrees (not radians), I get

sin 359.9999 = -1.745329251...e-6

The ellipsis (...) means that the display shows a lot more digits, but there's no reason to write them all down here. Rounding off this quantity to four significant figures and putting it into standard power-of-10 notation, I get -1.745 x 10-6. The correct choice is C.

8. Again using my computer's calculator in the scientific mode and setting it to work in degrees, I get

tan 270.0003 = -190985.9317...

Rewriting this large negative number in power-of-10 notation and rounding off to four significant figures yields -1.910 x 105. The correct choice is C.

9. Once again using my computer's calculator in the scientific mode, I get

tan 269.9997 = 190985.9317...

Rewriting this large positive number in power-of-10 notation and rounding off to four significant figures yields 1.910 x 105. The correct choice is D.

10. To find the cotangent of a quantity when a calculator doesn't have a cotangent key (as mine does not), find the tangent first, and then simply take the reciprocal. I set my calculator in the scientific mode and got

tan 269.9997 = 190985.9317...

leaving in all of the display's digits to avoid any rounding error that might occur when I take the reciprocal. Then I hit the reciprocal key to determine the fact that

cot 269.9997 = 1 / (tan 269.9997)
= 5.235987756...e-6

which translates and rounds off to 5.236 x 10-6. The answer is A.

Supplemental Note: I chose the values for the angles in Questions 7 through 10 deliberately, so that you can see what happens when the circular functions get close to "zero points" (such as sin 360 or cot 270) or "blowup points" (such as tan 270). You might want to test some values that get even closer to "zero points" or "blowup points," such as the following:

sin 359.999999

tan 270.000001

tan 269.999999

cot 270.000001

cot 269.999999

And then, after that, maybe these:

sin 359.99999999

tan 270.00000001

tan 269.99999999

cot 270.00000001

cot 269.99999999