**Introduction to Trigonometry and More abut Circles **

*Trigonometry* (or “trig”) involves angles and distances. Trigonometry scares some people because of the Greek symbology, but the rules are clear-cut. Once you can get used to the idea of Greek letters representing angles, and if you’re willing to draw diagrams and use a calculator, basic trigonometry loses most of its fear-inspiring qualities. You might even find yourself having fun with it.

**More about Circles**

Circles are defined by equations in which either *x* or *y* can be considered the dependent variable. We’ve already looked at the graphs of some simple circles in the *xy* -plane. Let’s look more closely at this special species of geometric figure.

**General Equation of a Circle**

The equation in the *xy* -plane that represents a circle depends on two things: the radius of the circle, and the location of its center point.

Suppose *r* is the radius of a circle, expressed in arbitrary units. Imagine that the center point of the circle in Cartesian coordinates is located at the point *x* = *a* and *y* = *b* , represented by the ordered pair ( *a* , *b* ). Then the equation of that circle is:

( *x* – *a* ) ^{2} + ( *y* – *b* ) ^{2} = *r* ^{2}

If the center of the circle happens to be at the origin, that is, at (0,0) on the coordinate plane, then the general equation is simpler:

*x* ^{2} + *y* ^{2} = *r* ^{2}

**The Unit Circle**

Consider a circle in rectangular coordinates with the following equation:

*x* ^{2} + *y* ^{2} = 1

This is called the *unit circle* because its radius is 1 unit, and it is centered at the origin (0,0). This circle gives us a good way to define the common trigonometric functions, which are sometimes called *circular functions* .

**It's Greek To Us**

In trigonometry, mathematicians and scientists have acquired the habit of using Greek letters to represent angles. The most common symbol for this purpose is an italic, lowercase Greek theta (pronounced “THAY-tuh”). It looks like an italic numeral *0* with a horizontal line through it ( *θ* ).

When writing about two different angles, a second Greek letter is used along with *θ* . Most often, it is the italic, lowercase letter phi (pronounced “FIE” or “FEE”). This character looks like an italic lowercase English letter *o* with a forward slash through it ( *Φ* ).

Sometimes the italic, lowercase Greek alpha (“AL-fuh”), beta (“BAY-tuh”), and gamma (“GAM-uh”) are used to represent angles. These, respectively, look like this: *α, β, γ* . When things get messy and there are a lot of angles to talk about, numeric subscripts are sometimes used with Greek letters, so don’t be surprised if you see text in which angles are denoted *θ* _{1} , *θ* _{2} , *θ* _{3} , and so on.

**Angular Units**

There are two main units by which the measures of angles in a flat plane can be specified: the radian and the degree. The radian was defined in the previous chapter. A quarter circle is π/2 rad, a half circle is π rad, and three quarters of a circle is 3π/2 rad. Mathematicians prefer to use the radian when working with trigonometric functions, and the abbreviation “rad” is left out. So if you see something like *θ* _{1} = π/4, you know the angle *θ* _{1} is expressed in radians.

The angular degree (°), also called the *degree of arc* , is the unit of angular measure most familiar to lay people. One degree (1°) is 1/360 of a full circle. An angle of 90° represents a quarter circle, 180° represents a half circle, 270° represents three quarters of a circle, and 360° represents a full circle. A right angle has a measure of 90°, an acute angle has a measure of more than 0° but less than 90°, and an obtuse angle has a measure of more than 90° but less than 180°.

To denote the measures of tiny angles, or to precisely denote the measures of angles in general, smaller units are used. One *minute of arc* or *arc minute* , symbolized by an apostrophe or accent (′) or abbreviated as m or min, is 1/60 of a degree. One *second of arc* or *arc second* , symbolized by a closing quotation mark (″) or abbreviated as s or sec, is 1/60 of an arc minute or 1/3600 of a degree. An example of an angle in this notation is 30° 15′ 0″, which is read as “30degrees, 15minutes, 0seconds.”

Alternatively, fractions of a degree can be denoted in decimal form. You might see, for example, 30.25°. This is the same as 30° 15′ 0″. Decimal fractions of degrees are easier to work with than the minute/second scheme when angles must be added and subtracted, or when using a conventional calculator to work out trigonometry problems. Nevertheless, the minute/second system, like the English system of measurements, remains in widespread use.

**Trigonometry—More about Circles Practice Problems**

**Practice 1**

A text discussion tells you that *θ* _{1} = π/4. What is the measure of *θ* _{1} in degrees?

**Solution 1**

There are 2π rad in a full circle of 360°. The value π/4 is equal to 1/8 of 2π. Therefore, the angle *θ* _{1} is 1/8 of a full circle, or 45°.

**Practice 2**

Suppose your town is listed in an almanac as being at 40° 20′ north latitude and 93° 48′ west longitude. What are these values in decimal form? Express your answers to two decimal places.

**Solution 2**

There are 60 minutes of arc in one degree. To calculate the latitude, note that 20′ = (20/60)° = 0.33°; that means the latitude is 40.33° north. To calculate the longitude, note that 48′ = (48/60)° = 0.80°; that means the longitude is 93.80° west.

Find practice problems and solutions for these concepts at: Trigonometry Practice Test.

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