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:
Now the trig. ratios can be defined for any acute angle as follows:
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) Then sin = y/r, cos = x/r, tan = y/x, etc. Now we can extend the definition of trig. ratios to any angles. These have the same value for the trig. functions except for possibly a change of sign. (The comparable acute angle is known as the reference angle.) For instance in the triangle:
sin is the same as sin but cos = -cos .
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
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.
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 symbol for degrees is ° , for minutes is ', and for seconds is ".
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.
It is seldom necessary to convert from minutes and seconds to decimals or vice versa; however, if you use the function tables of many trade manuals, it is necessary. Some tables show the fractions of degrees in minutes and seconds (DMS) rather than decimals (DD). In order to calculate using the different function tables, you must be able to convert the fractions to either format.
Converting Degrees, Minutes, & Seconds to Degrees & Decimals
To convert degrees, minutes, and seconds (DMS) to degrees and decimals of a degree (DD):
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.
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.
Web page design by Sarah Seastone for the Geometry Forum 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.
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").
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.
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° .
(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.
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.
(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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The Math Forum is a research and educational enterprise of the Drexel School of Education.
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.