Lecture 14: Trigonometric Functions Angle measure
Lecture 14: Trigonometric Functions
Angle measure
An angle AOB consists of two rays R1 and R2 with a common vertex O. We can think of the angle as
a rotation of the side R2 about the point O with R1 remaining fixed. R1 is called the initial side of the
angle and R2 is called the terminal side of the angle. If the rotation is counterclockwise, the angle is
positive as shown on the left below. If the rotation is clockwise, the angle is negative as shown on the
right below.
R1
O
A
¦¨
B
R2
R2
B
¦¨
O
R1
A
Angle Measure
There are two commonly used measures for angles, degrees and radians. In calculus, we will almost
exclusively use radian measure since the trigonometric functions are defined in terms of this measure.
Radian Measure If a circle of radius 1 is drawn with the vertex of an angle at its center, then the
measure of this (positive) angle in radians is the length of the arc that subtends the the angle (see
the picture below).
r=1
Radian measure of ¦¨
¦¨
The circumference of a circle of radius 1 is 2¦Ð, so a complete revolution has measure 2¦Ð. A straight
angle has measure ¦Ð and a right angle has measure ¦Ð/2.
1
1 rad
O
r=1
2 rad
¦¨
r=1
¦¨
¦¨
r=1
O
O
- ¦°?2 rad
¦° rad
¦°?2 rad
¦¨
¦° rad
2
¦¨
¦¨
r=1
r=1
r=1
O
O
O
Degrees The degree measure of a full circle is 360o . Therefore we have that 360o = 2¦Ð radians. This
gives us our conversion formulas.
2¦Ð
¦Ð
1o =
=
radians,
360
180
¦Ð
.
Therefore to convert from degrees to radians, we multiply by
180
We also have
360
180
1 radian =
=
degrees.
2¦Ð
¦Ð
180
Thus to convert from radians to degrees we must multiply by
.
¦Ð
Example Convert the following angles given in degrees to their radian measure:
45o ,
60o ,
30o ,
180o .
Example Convert the following angles given in radians to their measure in degrees:
¦Ð
10
?¦Ð
6
3¦Ð
4
2
5¦Ð
.
6
Angles in Standard Position and Co-terminal angles An angle is said to be in standard position
if it is drawn so that its initial side is the positive x-axis and its vertex is the origin. Two different
angles may have terminal sides which coincide, in which case we call the angles coterminal. This may
happen if one of the angles makes more than one revolution about the vertex or if the angles involve
revolutions in opposite directions. The angles 1 rad, 2 rad, ¦Ð rad, ¦Ð/2 rad, ?¦Ð/2 rad and 2¦Ð rad shown
above are all in standard position.
3¦Ð
¦Ð
5¦Ð
Coterminal Angles We see below that the angles
and ? are co-terminal. Also the angles
2
2
2
¦Ð
and are co-terminal.
2
3¦°?2 rad
1.0
5¦°?2 rad
0.5
O
r=1
-1.0
¦°?2 rad
r=1
-0.5
O
0.5
1.0
-0.5
- ¦°?2 rad
-1.0
Trigonometry of Right Triangles Recall the definition of sin ¦È, cos ¦È and tan ¦È from trigonometry
of right angles. If ¦È is the angle shown in the right triangle below, we have
sin ¦È =
y
length of side opposite ¦È
x
length of side adjacent to ¦È
=
, cos ¦È = =
,
r
hypotenuse
r
hypotenuse
tan ¦È =
length of side opposite ¦È
y
=
.
x
length of side adjacent to ¦È
r=
¦¨
x2 + y2
x
3
y
Example Find cos ¦È, sin ¦È and tan ¦È where ¦È is shown in the diagram below:
1.5
¦¨
1
We also have three further trigonometric functions which are often used and referred to in calculus.
sec x =
1
,
cos x
cscx =
1
sin x
cot x =
1
.
tan x
Example Find csc ¦È, sec ¦È and cot ¦È for the angle ¦È shown in the diagram in the previous example.
4
Special Triangles We can derive the cosine, sine and tangent of some basic angles by considering some
special right triangles. By drawing a square with sides of length 1, and its diagonal as shown, we can
use the Pythagorean theorem to find the length of the diagonal.
1
¦¨
1
Example Use the triangle above to determine:
sin(45o ) = sin(¦Ð/4),
cos(¦Ð/4),
tan(¦Ð/4),
sec(¦Ð/4),
csc(¦Ð/4),
cot(¦Ð/4).
By drawing and equilateral triangle with sides of length 2, we know that all angles are equal and add to
180o or ¦° radians. Thus each angle is a 60o angle or has a measure of ¦Ð/3 radians. From our Euclidean
geometry, we know that the line which bisects the upper angle in this (isosceles) triangle, also bisects
the base. Using the Pythagorean theorem we can find the length of the bisector and thus the sine,
cosine and tangent of angles of size 30o = ¦Ð6 and 60o = ¦Ð3 .
2
2
¦°?3
1
1
Example Use the triangle above to determine:
sin(¦Ð/6),
cos(¦Ð/6),
tan(¦Ð/6),
sec(¦Ð/6),
csc(¦Ð/6),
cot(¦Ð/6).
sin(¦Ð/3),
cos(¦Ð/3),
tan(¦Ð/3),
sec(¦Ð/3),
csc(¦Ð/3),
cot(¦Ð/3).
and
5
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