Similar Triangles, Right Triangles, And The Definition Of The Sine, Cosine And Tangent Functions Of Angles Of A Right Triangle Worksheets With Answer Key Page 3
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43Example: Consider a circle with radius 4. What is the area of a sector determined by a 45° angle?
π
°
=
Solution: First convert to radians:
45
.
4
π
θ
=
And so with
we have
4
π
( )
⋅
2
4
4
π
=
=
A
2
.
2
Problems:
8. Find the area of a sector determined by a 196° angle in a circle of radius 6.
π
5
9. Find the area of a sector determined by
radians in a circle of radius 2.
3
THE TRIGONOMETRIC FUCTIONS
θ
For an angle
in standard position, let (x, y) be a point on its terminal side. By the Pythagorean
=
+
2
2
Theorem
r
x
y
.
The six trigonometric functions of θ are defined as
r
y
θ
=
θ
=
,
csc
sin
,
y
r
x
r
θ
θ
=
=
cos
,
sec
,
r
x
x
y
θ
=
θ
=
cot
.
tan
and
y
x
Figure T6
θ
For an acute angle
of a right triangle, as shown in Figure T7, the trigonometric functions can also be
θ
θ
defined in terms of the side opposite angle
, the side adjacent to angle
, and the hypotenuse. Thus,
θ
side
opposite
hypotenuse
θ
θ
=
=
sin
,
csc
,
θ
hypotenuse
side
opposite
θ
side
adjacent t
o
hypotenuse
θ
θ
=
=
cos
,
sec
,
θ
hypotenuse
side
adjacent t
o
θ
θ
side
opposite
side
adjacent t
o
θ
θ
tan =
=
and
cot
.
θ
θ
side
adjacent t
o
side
opposite
Figure T7
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