**Trigonometric Functions**

One of the easier ways to start understanding trigonometric functions is by picturing a right triangle. (Refer back to the triangles section to recall this.) Let be one of the acute angles. Then we will label the triangle as follows:

opposite

adjacent

opposite

sin =

cos =

tan =

hypotenuse

hypotenuse

adjacent

---

---

hypotenuse

hypotenuse

adjacent

csc =

sec =

cot =

opposite

adjacent

opposite

Note: adjacent

opposite

sin =

hypotenuse

adjacent

---

---

hypotenuse

hypotenuse

adjacent

csc =

adjacent

opposite

The ratios are the same for any right triangle with angle , since when a triangle has equal angles they are similar trianges.

Let's take a look at some examples

Now what if we have an angle greater than 90o (ie. an obtuse angle)? Well if we label the triangle like this:

(where r = SQRT(x2 + y2) by the Pythagorean Theorem)

From the note above, since the ratio is the same as long as the angles are the same, let's assume that r = 1. We can now place the triangle on the unit circle with O at the center of the circle and P as a point on the circle. Here's a diagram to see what we mean.

When we substitute r = 1 into the equations, we get sin = y and cos = x. Therefore, that means that the x-coordinate of the point P gives the value of cos and the y-coordinate of the point P gives the value of sin . From sin and cos , we can figure out the remaining trig. ratios. Here are some examples using trig. functions.

**Degrees**

**Degrees**are the units of measurement for angles.

**There are 360 degrees in any circle, and one**

**degree is equal to 1/360 of the complete****rotation of a circle.**360 may seem to be an unusual number to use, but this part

of math was developed in the ancient Middle East. During

that era, the calendar was based on 360 days in a year, and

one degree was equal to one day.

**Fractions of Degrees**

There are two methods of expressing fractions of degrees.

**The first method**divides each degree into**60 minutes**(1° = 60'), then each minute into**60 seconds**(1' = 60").

**37° 42' 17"**

The symbol for degrees is ° , for minutes is ', and for seconds is ".

**The second method**states the fraction as a decimal of a degree. This is the method we will use.

**37° 42' 17"**expressed as

**37.7047°**.

Most scientific calculators can display degrees both ways. The key for degrees on my calculator looks like

**° ' "**, but the key on another brand may look like

**DMS**. You will need to refer to your calculator manual to determine the correct keys for degrees. Most calculators display answers in the form of degrees and a decimal of a degree.

**Converting Degrees, Minutes, & Seconds to Degrees & Decimals**

To convert degrees, minutes, and seconds (DMS) to degrees and decimals of a degree (DD):

**First:**Convert the seconds to a fraction.

Since there are 60 seconds in each minute, 37° 42' 17" can be expressed as

37° 42 17/60'. Convert to 37° 42.2833'.**Second:**Convert the minutes to a fraction.

Since there are 60 minutes in each degree, 37° 42.2833' can be expressed as

37 42.2833/60° . Convert to 37.7047° .

**Degree practice 1:**Convert these DMS to the DD form. Round off to four decimal places.

(1) 89° 11' 15" (5) 42° 24' 53" (2) 12° 15' 0" (6) 38° 42' 25" (3) 33° 30' (7) 29° 30' 30" (4) 71° 0' 30" (8) 0° 49' 49"

**Answers.**

**Converting Degrees & Decimals to Degrees, Minutes, & Seconds**

To convert degrees and decimals of degrees (DD) to degrees, minutes, and seconds (DMS), referse the previous process.

**First:**Subtract the whole degrees. Convert the fraction to minutes. Multiply the decimal of a degree by 60 (the number of minutes in a degree). The whole number of the answer is the whole minutes.**Second:**Subtract the whole minutes from the answer.**Third:**Convert the decimal number remaining (from minutes) to seconds. Multiply the decimal by 60 (the number of seconds in a minute). The whole number of the answer is the whole seconds.**Fourth:**If there is a decimal remaining, write that down as the decimal of a second.

**DMS is stated as 5° 14' 4.416"**

**Degree practice 2:**Convert these DD to the DMS form.

(1) 75.25° (5) 13.12345° (2) 43.375° (6) 21.5° (3) 9.5625° (7) 59.7892° (4) 33.9645° (8) 65.1836°

**Answers.**

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Here are the sine and cosine curves. Notice that the graphs repeat themselves every 2 . (In other words, the graphs have a period of 2 .)

Since the cscx and secx are the same as 1/sinx and 1/cosx, respectively, we would assume that the period would be the same. From the diagram we also note that we get the real numbers which are not defined by these functions.

The remaining two are slightly different. They only have period . They too have the undefined numbers for the function.