TRIGONOMETRIC RATIOS OF ANY ANGLE
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to ingrid.stewart@csn.edu.
Please read this lecture carefully and several times! Many students have a hard time understanding these concepts if they don't spend enough time "wrestling" with them.
In a previous lecture we defined the trigonometric ratios as follows using a right triangle:
In this lecture we will extend the definition of the trigonometric ratios to include angles that are not found in a right triangle.
Let's first place a right triangle into the Cartesian Coordinate System so that the right angle lies opposite the Origin.
Please note that we will NOT call the length of the hypotenuse c, instead we will call it r. Furthermore, observe that the length of leg a and the length of leg b produce the point (a, b) in the coordinate system. Angle is in between side a and side r.
Given the triangle above, the numeric values of the six trigonometric ratios of angle are now defined as follows. Please understand that angle can be a radian or degree measure!
where .
NOTE: YOU MUST MEMORIZE THESE DEFINITIONS OF THE TRIGONOMETRIC RATIOS!
From now on, we will NOT consider a and b to be lengths of the legs of a right triangle as has been done in a previous lecture. Instead, a and b are now considered to be positive or negative numbers depending on the location to the terminal side of the angle in the coordinate system.
Example:
Let's copy the following triangle into each quadrant of a coordinate system. Then we will find the numeric values of the cosine ratio for angles with their vertex next to the origin.
Remember that and
Note that r = 2 in this example!
Then
cos
(60)
=
(QI Angle)
cos
(120)
= (QII
Angle)
cos
(240)
=
(QIII Angle)
cos
(300)
=
(QVI Angle)
cos (60)
=
(QVI Angle) cos (120)
=
(QIII Angle)
cos (240)
= (QII
Angle)
cos (300)
=
(QI
Angle)
Please note that in the picture above, the numeric value of the cosine ratio is positive since the point has a positive a and b value and the distance r is always positive.
On the other hand, the point in the second quadrant has a negative a and a positive b value. Therefore, the cosine ratio has a negative numeric value.
Likewise, the point in the third quadrant has both a negative a and b. Therefore, the cosine ratio has again a negative numeric value.
Finally, the point in the fourth quadrant has a positive a and a negative b. Therefore, the cosine ratio has a positive numeric value.
This pattern of negative and positive numeric values as discussed above can be generalized to all trigonometric ratios and is summarized in the following table. Try it with the other 5 trigonometric ratios and different angles!
Terminal side of the angle is in Quadrant I All trigonometric ratios have positive numeric values. Terminal side of the angle is in Quadrant II Sine and cosecant ratios have positive numeric values, all others have negative values. Terminal side of the angle is in Quadrant III Tangent and cotangent ratios have positive numeric values, all others have negative values. Terminal side of the angle is in Quadrant IV Cosine and secant ratios have positive numeric values, all others have negative values.
NOTE: YOU MUST MEMORIZE THIS SIGN PATTERN!
A handy memorization aid to remember this by is: All Students Take Calculus
Given the extended definition of the trigonometric ratios, we can now find EXACT numeric values of the special angle measures 0^{o}, 90^{o}, 180^{o}, 270^{o}, and 360^{o}.
NOTE: YOU MUST MEMORIZE THE FOLLOWING NUMERIC VALUES!
Coterminal Angles
Two angles are coterminal if they have the same initial and terminal sides. Any angle has infinitely many coterminal angles.
You can find an angle that is coterminal to a given angle by adding integer multiples of to the given angle. Remember that integers include the number 0 and positive and negative whole numbers!
For instance, the angles of magnitude 135^{o} and 585^{o }are coterminal.
135^{o} + 360^{o}(2) = 585^{o}. See picture below.
Reference Angles
The Reference Angle, let's call it (Greek letter ALPHA) from now on, is ALWAYS an acute (less than 90^{o}) and positive angle between the terminal side of an angle (Greek letter THETA) and the horizontal axis (x-axis) in a coordinate system.
If an angle is positive and between 0^{o} and 360^{o} and NOT quadrantal, you can calculate the magnitude of its Reference Angle as follows:
is a first-quadrant angle:
is a second-quadrant angle:
is a third-quadrant angle:
is a fourth-quadrant angle:
NOTE: YOU MUST MEMORIZE THE REFERENCE ANGLE CALCULATIONS!
For all other angles the following applies:
(1) Quadrantal Angles DO NOT have Reference Angles.
Reminder: When the terminal side of an angle lies along a coordinate axis, the angle is called a Quadrantal Angle. For example, 0^{o}, 90^{o}, 180^{o}, 270^{o}, 360^{o}, 450^{o}, etc. are considered Quadrantal Angles.
(2) Negative angles have the same Reference Angles as their positive counterparts.
(3) Trigonometric ratios of angles and trigonometric ratios of their Reference Angle have the same absolute numeric value!
(4) Coterminal Angles have the same Reference Angle
Find EXACT numeric values of trigonometric ratios of multiples of special angles WITH the calculator.
We will use the following example to illustrate the solution strategy. However, this is just one of many examples. Be sure to study the examples at the end of this lecture
EXAMPLE: Find the numeric value of sin 240^{o} with the calculator.
Step 1:
Find the Reference Angle of the given angle. Remember that angles with the same Reference Angle have the same absolute numeric value!!!!
Example continued: The Reference Angle is 240^{o} 180^{o} = 60^{o}
Use the calculator to find the numeric value of the given trigonometric expression.
Example continued: sin 240^{o} = 0.866 (rounded to 3 decimal places)
Using the Reference Angle as confirmation, we determine the EXACT value of the decimal approximated numeric value found in the previous step. Unless your calculator gives you EXACT numeric values, you must have memorized the decimal approximations of the numeric values of special angles found in the previous lecture!!!
Example continued: Using the Reference Angle as confirmation, we know that
0.866 . . . .... must be equal to . Therefore, sin 240^{o} =
Find EXACT numeric values of trigonometric ratios of multiples of special angles WITHOUT the calculator:
NOTE: YOU MUST BE ABLE TO DO THIS!
Now that we defined a Reference Angle, we can use it to find certain numeric values of trigonometric ratios WITHOUT using the calculator. Specifically, we can only do this with multiples of the special angles 0^{o}, 30^{o}, 45^{o}, 60^{o}, 90^{o}, 180^{o}, 270^{o}, and 360^{o}.
Procedure:
Example: Find the numeric value of sin 240^{o} without using the calculator:
Find the Reference Angle of the given angle. Remember that angles with the same Reference Angle have the same absolute numeric value!!!!
Example continued: The Reference Angle is 240^{o} 180^{o} = 60^{o}
Find the numeric value of the trigonometric ratio paired with this Reference Angle.
Example continued: We previously memorized that sin 60^{o} =
Use the memorization aid All Students Take Calculus to assign a positive or negative sign to the numeric value found in the previous step.
Example continued: We know that the angle 240^{o} is a third-quadrant angle. Therefore, the numeric value of sin 240^{o}^{ } must be negative using the memorization aid All Students Take Calculus.
Specifically, sin 240 =
Observe that !!!
Remember that trigonometric ratios of angles and trigonometric ratios of their Reference Angle have the same absolute numeric value!
So what's the point of not using a calculator? The reason is "speed". In any trigonometry or calculus course you are working a tremendous amount of problems and having a way to find answers without constantly asking your calculator is beneficial.
NOTE:
Some trigonometric ratios are paired with Quadrantal Angles, which DO NOT have a Reference Angle. In this case it is most important that you have memorized the values in the chart above. That's the only way you will be able to find certain EXACT numeric values WITHOUT using the calculator. See Examples 26, 27, and 28.
Example 01
Given a point in the coordinate system, find the EXACT numeric values of the six trigonometric ratios.Example 02
Given a point in the coordinate system, find the EXACT numeric values of the six trigonometric ratios.Example 03
Given a trigonometric ratio and its numeric value, find the location in the coordinate system of the terminal side of the angle.Examples 04 to 11
Find the Reference Angles of angles in degrees.Examples 12 to 17
Find the Reference Angles of angles in EXACT radians, that is, they contain the number .Examples 18 to 23
Find the Reference Angles of angles in decimal radians.Examples 24 to 33
Given trigonometric ratios with angles in degrees, find EXACT numeric values WITHOUT the calculator.Examples 34 to 43
Given trigonometric ratios with angles in radians, find EXACT numeric values WITHOUT the calculator.Examples 44 to 53
Given trigonometric ratios with angles in degrees and radians, find EXACT numeric values WITH the calculator.