Mathematics Notes by Success Tutorials: Trigonometry
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General
- trigonometry is the study of angles and how angles relate to each other
- in this section, we will study properties of angles called sine, cosine, tangent, cosecant, secant and cotangent
- in order to understand these properties of angles we first need to have a brief look at the unit circle
Unit Circle and Radians
- a unit circle is just a circle with a radius of 1 unit (i.e. 1 centimeter, 1 inch, etc.)
- having a radius of 1 unit is helpful because it allows angles within a circle to be expressed in a simple, invariable way:
C = 2(pi)r; r = radius
if r = 1, C = 2pi with no variability from differing radius measures pi is a number which comes up frequently in trigonometry
pi = 3.141592654...; it is a non-repeating, infinite decimal
- what we have done above is to use the formula for the circumference of a circle to determine the circumference of a unit circle
- the circumference of the unit circle is the radian measure used for all circles
- a circle contains 360 degrees, if you go all of the way around it, or 2pi radians
- we generally leave radian measures in terms of pi because this makes it easier to work with
- it is simpler to think of a circle as having 2pi radians than 6.283185308... radians
- if we work with a unit circle, we find that all unit circles have 2pi units in their circumference, with the only variability being whether the units may be expressed in metres, centimeters, inches, etc.
- radians are often expressed in pi units and are based on the unit circle
- all circles have 360 degrees, or 2pi radians
- we can now say that all 90 degree angles have pi/2 radians, 180 degree angles have pi radians, and so on
- this is how the unit circle simplifies the use of radians and trigonometry, eliminating the variability which would otherwise come from radii of different lengths
Converting from Degrees to Radians
- there are 360 degrees in a circle
- there are 2pi radians in a circle, this is determined using a unit circle (radius = 1)
- to convert from degrees to radians:
degrees/180 x pi = radians in angle being measured
- to convert from radians to degrees:
radians/pi x 180 = degrees in angle being measured
Sine, Cosine, Tangent, Cosecant, Secant and Cotangent
- now let's look at what is meant by "sine", and the other trigonometric functions
- the sine of an angle of a triangle is defined to be:
- the length of the side opposite to the angle divided by the length of the hypotenuse
- only 90 degree angles have hypotenuses so to use this definition you will need to work with a right triangle or be able to convert the angle you are looking at into an angle in a right triangle
- the above diagram shows what is meant by sine, cosine, tangent, cosecant, secant and cotangent
- all of these functions are defined on right triangles and will not work if right triangles are not being used (one angle in the triangle must equal 90 degrees)
- remember the word SOHCAHTOA to remember Sin(A)=Opp/Hyp, Cos(A)=Adj/Hyp and Tan(A)=Opp/Adj
- the SOHCAHTOA definition only works on right angle triangles and other means must be used to determine sine, cosine and tangent in non-right angle triangles, such as the Sine and Cosine Laws
Sine, Cosine, etc. and the Unit Circle
- we can find all of the trigonometric functions mentioned above on the unit circle, as well as in triangles
- let's take a second look at the previous diagram of the unit circle, adding a x-y co-ordinate system
- we can see that the sine of angle A is the opposite side/the hypotenuse, which equals y/r
- since the radius, r, is always equal to 1 in the unit circle, ratios are simplified
- we can clearly see why the ratios change sign from positive to negative as we move around a circle
Cast rule: A S T C
- to summarize: Sign by Quadrant: 1st 2nd 3rd 4th
1) sine A = opposite side/hypotenuse or y/r + + - -
2) cosine A = adjacent side/hypotenuse or x/r + - - +
3) tangent A = sine/cosine or opposite side/adjacent side or y/x + - + -
4) cosecant A = 1/sine or hypotenuse/opposite side or r/y + + - -
5) secant A = 1/cosine or hypotenuse/adjacent side or r/x + - - +
6) cotangent A = 1/tangent or adjacent side/opposite side or x/y + - + -
Examples
Q. What is the sine of a 0 degree angle?
A. If the angle is 0 degrees, the side opposite the angle is 0 units in length. Check this using the above diagram. If angle A is 0 degrees, how long must side a be? Or ask, what is the y-value of a point on the x-axis? The answer is that that side is 0 units in length.
Since sine = opposite side/hypotenuse = y/r, and the length of the opposite side is 0 units in length. the sine of a 0 degree angle is 0.
Q. What is the cosine of a 0 degree angle?
A. If the angle is 0 degrees, the adjacent side will be the same length as the hypotenuse.
since cosine = adjacent side/hypotenuse, and these two sides must be the same length, the cosine of a 0 degree angle is 1.
Q. What is the tangent of a 0 degree angle?
A. Tangent = opposite side/adjacent side, and as we have already seen the length of the opposite side is 0. This means that the tangent of a 0 degree angle is 0.
Q. What is the sine of a 90 degree angle?
A. This is the same as asking what the x-y co-ordinates of the point where the y-axis intersects the circle. The co-ordinates are (0,1), because the radius is 1 unit in length and the y-value is the same magnitude as the radius at this point. Since sine = y/r, the sine of a 90 degree angle is 1/1, or 1.
Q. What are the sine and cosine of a 45 degree angle?
A. The co-ordinates of the point on the unit circle where a line at 45 degrees intersects the circle are (1/sq.rt.2, 1/sq.rt.2). This can be determined by constructing a right triangle with a hypotenuse of length 1 and two other sides of equal length. Using the Pythagorean Theorem we find that a2 and b2 must each be 1/2, since they must add up to equal c2, which is 1.
This means the sine of a 45 degree angle is y/r, or (1/sq.rt.2)/1, or 1/sq.rt.2.
The cosine of a 45 degree angle is x/r, or (1/sq.rt.2)/1, or 1/sq.rt.2.
Q. What is the sine and cosine of a 135 degree angle?
A. The co-ordinates of this point on the circle are (-1/sq.rt.2, 1/sq.rt.2). This has been shown above using the Pythagorean Theorem, since a 135 degree angle is just an angle which is 45 degrees past the 180 degree line. The difference is that the x co-ordinate is in the negative direction.
This means the sine of a 135 degree angle is y/r, or (1/sq.rt.2)/1, or 1/sq.rt.2.
Since cosine is x/r, the cosine of a 135 degree angle is (-1/sq.rt.2)/1, or -1/sq.rt.2.
Summary Table of Values for Trigonometric Functions
Angle A= 0pi pi/2 pi 3pi/2 2pi radians
0
Sine A 0 1 0 -1 0
Cosine
A 1 0 -1 0 1
Tangent
A 0 -- 0 -- 0
Cosecant A -- 1 -- -1 --
Secant A 1 -- -1 -- 1
Cotangent A -- 0 -- 0 --
Note: some of the above functions are undefined at the angles specified (denoted --). This is because those functions involve dividing by zero at that angle.
i.e. Tangent A at 90o is undefined because tangent A = sine A/cosine A = 1/0
Trigonometric Identities (for Angles A and B)
sin A = 1/(csc A)
cos A = 1(sec A)
tan A = 1/(cot A)
tan A = sin A/(cos A)
cot A = cos A/(sin A)
tan A = sec A/(csc A)
sec A = cos A + tan A (sin A)
1/(csc A - sin A) = tan A (sec A)
Pythagorean Identities
sin2 A + cos2 A = 1
tan2 A + 1 = sec2 A
cot2 A + 1 = csc2 A
Negative Arc Identities
sin(-A) = -sin A
cos(-A) = cos A
tan(-A) = -tan A
cot(-A) = -cot A
sec(-A) = sec A
csc(-A) = -csc A
Cofunction Identities
sin(pi/2 - A) = cos A
cos(pi/2 - A) = sin A
tan(pi/2 - A) = cot A
cot(pi/2 - A) = tan A
sec(pi/2 - A) = csc A
csc(pi/2 - A) = sec A
Trigonometric Identities (for Angles A and B) continued...
Sum and Difference Identities (Compound Angle Identities)
sin(A - B) = sin A(cos B) - cos A(sin B)
sin(A + B) = sin A(cos B) + cos A(sin B)
cos(A - B) = cos A(cos B) + sin A(sin B)
cos(A + B) = cos A(cos B) - sin A(sin B)
tan(A - B) = (tan A - tan B)/(1 + tan A(tan B))
tan(A + B) = (tan A + tan B)/(1 - tan A(tan B))
Double-Angle Identities
sin(2A) = 2sin A(cos A)
cos(2A) = 2cos2 A - 1
and cos(2A) = 1 - 2sin2 A
and cos(2A) = cos2 A - sin2 A
tan(2A) = (2tan A)/(1 - tan2 A)
Half-Angle Identities
cos(1/2 A) = +/- sq.rt.(1/2(cos A + 1)) also cos2(1/2 A) = 1/2(cos A + 1)
sin(1/2 A) = +/- sq.rt.(1/2(1 - cos A)) also sin2(1/2 A) = 1/2(1 - cos A)
tan(1/2 A) = +/- sq.rt.((1 - cos A)/(1 + cos A)) also tan2(1/2 A) = (1 - cos A)/(1 + cos A)
tan(1/2 A) = sin A/(1 + cos A)
tan(1/2 A) = (1 - cos A)/(sin A)
Two Important Laws, Based on the Following Diagram
1. Law of Sines
sin A / a = sin B / b = sin C / c
The Law of Sines can be used for all types of triangles, not just right triangles.
2. Law of Cosines
a2 = b2 + c2 - 2bc(cos A)
or b2 = a2 + c2 - 2ac(cos B)
or c2 = a2 + b2 - 2ab(cos C)
There are three forms of the Law of Cosines, note the pattern.
The Law of Cosines can be used for all types of triangles, not just right triangles.
Opdateret d. 23/2/01
