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Sometimes, for convenience, we assume a circle of radius r = 1, called a unit circle, when defining or evaluating the values of the trigonometric functions.
Typology: Summaries
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Radian Measure
Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by:
θ = s r
This defintion of an angle in radian measure is independent of the radius of the circle. A larger circle with a longer radius, R, will also sustain a longer arc, S,
and the ratio,
R will be the same as^
s r. In other words,^ θ^ =^
s r =^
θ r
s θ R
For a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is
θ = πr r = π
Since a (semi-circle) straight angle has measure 180◦, π radian is equivalent to 180 ◦.
Given an angle measurement in degree, multiply that number by π 180 ◦^ to find the radian measure.
Given an angle measurement in radian, multiply that number by
π to find the degree measure.
Example: What is 55◦^ in radian?
Ans: 55◦^ · π 180 ◦^
π =
π ≈ 0. 31 π ≈ 0. 96
Example: What is 45◦^ in radian?
Ans: 45◦^ · π 180 ◦^
π =^1 4 π = π 4
Unless a decimal approximation is desired, we should always leave the number
in exact format. That is,
π or 11 π 36 is the most desirable way of writing the above angle measure in radian.
You should note that radian measure is a (real) number, and is more canon- ical than degree measure when used in working with mathematical functions of real numbers.
If an angle measurement is written without the ◦ symblo in the upper right, it is a radian measure. If the ◦ is present, it is a degree measure.
Distingush the difference between 1◦^ and 1 (radian).
With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive side of the x−axis. If we obtained the other side (Called the Terminal Side) of the angle via a counter-clockwise rotation, we have a positive angle. If the terminal side of the angle is obtained via a clockwise rotation, we have a negative angle.
Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
E.g. The terminal side of π 4 is in the first quadrant.
E.g. The terminal side of − π 6 is in the fourth quadrant.
E.g. The terminal side of 2 π 3 is in the second quadrant.
E.g. The terminal side of 4 π 3 is in the third quadrant.
E.g. The terminal side of^16 π 7 is in the first quadrant.
E.g. The terminal side of π is the negative x−axis.
E.g. The terminal side of − π 2 is the negative y−axis.
Two angles are co-terminal angles if they have the same terminal side.
E.g. The two angles π 3 and − 5 π 3 are co-terminal.
Notice that if θ is any angle, then the angles θ + 2π, θ + 4π, θ + 6π, θ + 8π, · · · are all co-terminal angles
Given a Circle with radius r centered at the origin (The equation of this circle is x^2 + y^2 = r^2 ), we define the terminal point of an angle θ to be the point of intersection of the circle with the terminal side of θ.
E.g. The terminal point P of θ = π 6 is in the first quadrant.
If the radius of the circle is r = 1, then the coordinate of this terminal point P
is
If the radius of the circle is r = 3, then the coordinate of this terminal point P
is
E.g. The terminal point of θ = π 2 is on the positive side of the y−axis. If the radius of the circle is r = 1, then the coordinate of the terminal point of θ is (0, 1).
Note:
Any co-terminal angle has the same terminal point.
If θ is an angle with terminal point P , then any angle of the form θ + 2π, θ + 4 π, θ + 6π, θ + 8π, · · · all have P as the terminal point.
To find the coordinate of the terminal point, we need to know the radius of the circle and the measurement of the angle.
Given an angle θ in standard position, the reference angle of θ is the acute angle that the terminal side of θ makes with the x−axis. To find the coordinate of the terminal point, it is most often easier to consider the length of the sides of the right triangle formed by the terminal side of θ and the x−axis with the reference angle of θ being one of the interior angles.
In the above picture, θ is the angle in standard position and β is the reference angle. Note that the reference angle must be angle between 0 to π 2 (0 to 90◦).
Example: The reference angle of a 120◦^ angle is a 60◦^ angle.
Example: The reference angle of a − 235 ◦^ angle is a 55◦^ angle.
Example: For θ = π 3 , find the values of the six tri functions of θ.
Notice that π 3 = 60◦. Assuming a circle of radius 1, the terminal point P of θ
has coordinate
, so the values of the trig functions would be:
cos θ = x r =
1 2 1 =
2 sin^ θ^ =^
y r =
√ 3 2 1 =
tan θ = y x
√ 3 2 1 2
3 cot θ = x y
1 √^2 3 2
sec θ = r x
2
= 2 csc θ = r y
2
Example:
Assuming a unit circle, if θ = 5 π 6 = 150
◦, its reference angle is π 6 = 30
◦, and the
terminal point P of θ has coordinate
. Therefore,
sin
5 π 6
cos
5 π 6
tan
5 π 6
1 2 −
√ 3 2
Example:
x
y θ = 7 π 6
r = 1
P =
( −
√ 3 2 ,^ −^
1 2
)
π 1 6 2
If we use a unit circle, when θ = 7 π 6 (210◦), then its reference angle is π 6
and the terminal point P of θ has coordinate
. Therefore,
sin
7 π 6
cos
7 π 6
tan
7 π 6
√ 3 2
Example:
x
y
θ =^2 π 3
r = 1
P =
( − 12 ,
√ 3 2
)
π 3
If we use a unit circle, when θ = 2 π 3 (120◦), then its reference angle is π 3
and the terminal point P of θ has coordinate
. Therefore,
sin
2 π 3
cos
2 π 3
tan
2 π 3
√ 3 2 −^12
Example:
x
y P = (0, 1)
θ = π 2
If we use a unit circle, when θ = π 2 (90◦), then its terminal point lies on the
y−axis and has coordinate (0, 1). Therefore,
sin
(π 2
cos
(π 2
tan
(π 2
= undefined.
While in all the examples we did we assumed a circle of radius 1, it is important for you to know that, in order to find the value of the trig functions on an angle θ, we can use a circle of any radius. We used 1 out of convenience, but our answer would be the same if we had used a circle of radius 2, or 0.5, or any other positive number.
Example: The terminal point of angle θ has coordinate (3, −1), find the value of the six trigonometric functions of θ.
x
y
θ r =
P = (3, −1)
In this example, it would be more convenient to solve the problem if we use a circle of radius
10 (why?), then according to the picture,
sin θ = y r
cos θ = x r
tan θ = y x
Notice that in solving this problem, we do not need to know the value of θ. In fact, we do not even know if θ is positive or negative. The only information we have (and it is the only information we need) is the coordinate of the terminal point of θ.
Fundamental Properties of The Trigonometri Functions:
sin and csc are reciprocal functions of each other, that is:
csc(x) =
sin(x)
cos and sec are reciprocal functions of each other:
sec(x) =
cos(x)
tan and cot are reciprocal functions of each other:
cot(x) =
tan(x)
In addition, tan is the quotient of sin and cos:
tan(x) = sin(x) cos(x)
cot(x) = cos(x) sin(x)
The following equation can be derived from the pythagorean theorem, and is called the Pythagorean Identity: For all real numbers x, we have:
sin^2 (x) + cos^2 (x) = 1
Note: sin^2 (x) means (sin x)^2. In general, to represent (sin x)n, we write sinn(x). This notation applies to other tri functions too.
More properties of the tri functions:
sin (and its reciprocal, csc), is an odd function, that is,
sin(−x) = − sin(x) for all real numbers x
cos (and its reciprocal, sec), is an even function, that is,
cos(−x) = cos(x) for all real numbers x
The product (and quotient) of an odd function with an even function is odd, so tan and cot are both odd functions.
