**1. Order of Arithmetic Operations**- Certain arithmetic operations take precedence over others. In completing problems with a series of operations the following guidelines apply:
- a. Addition or subtraction may occur in any order.
*Example:*4 + 8 − 7 + 3 = 8 or 8 + 3 + 4 − 7 = 8

- b. Multiplication or division must be completed before addition
or subtraction.
*Example:*48 ÷ 6 + 2 = 10*Example:*4 + (2/3)(1/2) = 4 1/3

- c. Any quantity above a division line, under a division line
or a radical sign , or within parentheses or brackets must be
treated as one number.

**2. Fractions, Decimals, and Percents**- a. To add (or subtract) fractions, the denominator in
each term must be the same. (Choose the lowest common denominator
for each term. Multiply each term by the common denominator and
then add [or subtract].)
- (lowest common denominator = 12)

*Solution:*- (lowest common denominator =
*xc*)

- (lowest common denominator =
*Solution:*

- b. To multiply fractions, multiply the numerators by each
other and the denominators by each other.
- c. To divide fractions, invert the divisor and multiply.
- d. To convert a fraction to a percentage divide the numerator
by the denominator and multiply by 100.
*Note:*To convert a percentage to a decimal move the decimal point two places to the left.

- e. When
*dividing*by a decimal divide by the integer and add sufficient zeros to move the decimal point the appropriate number of digits to the*right.*- (appropriate number of digits to right = 2)
- When
*multiplying*by a decimal multiply the integer and add enough zeros to move the decimal point the appropriate number of digits to the*left.* - (appropriate number of digits to left = 3)

- f. Decimals may be expressed as positive or negative powers
of 10:

- a. To add (or subtract) fractions, the denominator in
each term must be the same. (Choose the lowest common denominator
for each term. Multiply each term by the common denominator and
then add [or subtract].)
**3. Proportions, Formulas, and Equations**- The location of values in proportions, equations, or formulas may be shifted provided that whatever addition, subtraction, multiplication, or division is performed on one side of the equation is also performed on the other side.

**4. Right Triangles and Trigonometric Equations**- a. In a right triangle one angle always equals 90°. The other two angles will always be acute angles and the sum of these two angles will be 90° since the sum of the angles in any triangle is 180°.
- b. In a right triangle the
*sides*are related to each other so that the square of the longest side or*hypotenuse*(*c*) is equal to the sum of the squares of the two sides:*c*2 =*a*2 +*b*2. This is the Pythagorean theorem. - c. In triangle
*ABC,*side*a*is called the side opposite angle*A,*side*b*is opposite angle*B,*and the hypotenuse,*c,*is opposite the right angle. Side*b*is named the side*adjacent*to angle*A*and side*a*is the side adjacent to angle*B.* - d.
*Trigonometric functions*are ratios between the sides of a right triangle and are determined by the value of one of the acute angles. There are six trigonometric functions—the sine, cosine, tangent, cotangent, secant, and cosecant—but it will be necessary to consider only the first four here.- In Δ
*ABC*the ratio between the side opposite one of the acute angles and the hypotenuse is called the*sine*of the angle. For angle*A*it would be written as - The
*cosine*expresses the ratio between the side adjacent and the hypotenuse. For angle - The
*tangent*and*cotangent*represent ratios between the two sides of the - A glance at these values shows that sin
*A*= cos*B,*and that tan*A*= cot*B.* - As can be seen from studying these ratios, two functions may
have the same ratio. For instance, the sin
*A*= cos*B*and the tan*A*= cot*B.* - In general terms these trigonometric functions are expressed as follows:
- *θ (Greek letter, theta) is the symbol for angle.

- In Δ
- e. Value of trigonometric functions may be obtained from tables
of trigonometric functions (see Appendix E) or from handheld calculators
with trigonometric function capability.
*Example:*sin 60° = .8660- cos 30° = .8660
- tan 22° = .4040
- cot 68° = .4040

- Tables of trigonometric functions usually go up to 90°.
Angles greater than 90° may be handled as follows:
- (1) Functions of angles greater than 90° but
less than 180° are the same as functions of an
angle equal to 180° minus the angle in question.
All functions of angles in this range are negative except the sine.
*Example:*sin 120° = sin 60°- tan 150° = –tan 30°

- (2) Functions of angles greater than 180° but
less than 270° are the same as functions of an
angle equal to 270° minus the angle in question.
Functions of angles in this range are negative except for the tan
and cot.
*Example:*cot 220° = –cos 50°- tan 195° = tan 75°

- (3) Functions of angles greater than 270° but
less than 360° are the same as functions of an
angle equal to 360° minus the angle in question.
All functions of angles in this range are negative except the cosine.
*Example:*cot 300° = –cot 60°- sin 330° = –sin 30°

- (1) Functions of angles greater than 90° but
less than 180° are the same as functions of an
angle equal to 180° minus the angle in question.
All functions of angles in this range are negative except the sine.

- f. Through the use of trigonometric functions, it is possible
to determine the values of all components of a triangle when the
values of
*one side and one angle*or the values of*two sides*are known.*Example 1:*In triangle*ABC,*angle*A*= 25° and the length of the hypotenuse is 15 m. Find the length of the other two sides.*Solution:**Example 2:*In triangle*ABC*the lengths of the sides are 3 cm and 5 cm. What is the length of the hypotenuse and the size of both acute angles?*Solution:**Note: c*may also be found using the Pythagorean theorem:*C*2 =*a*2 +*b*2

**5. Geometry of Circles**- a. The circumference of a circle is calculated using the formula
*C*= 2π*r*, where*C*is the circumference,*r*is the radius, and π (pi) is a constant value of 3.1416. Pi is the ratio that exists between the diameter of a circle and its circumference. - b. In making one complete turn about a circle the radius goes
through one revolution, 360° or 2π radians.
A radian is the angle subtended by an arc of a circle equal in length
to the radius. One radian equals
- or 57.3°. Some equivalents for these angular units of measure are as follows:
- (1) To convert degrees to revolutions divide by 360.
*Example:*1260° = 3.50 rev

- (2) To convert radians to revolutions divide by 6.28.
*Example:*15.75 radians = 2.51 rev

- (3) To convert degrees to radians divide by 57.3.
*Example:*360° = 6.28 radians

- (4) To convert revolutions to radians multiply by 6.28.
*Example:*2.3 rev = 14.44 radians

- (5) To convert revolutions to degrees multiply by 360.
*Example:*2.3 rev = 828°

- (6) To convert radians to degrees multiply by 57.3.
*Example:*7.6 radians = 435.5°

- a. The circumference of a circle is calculated using the formula

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