FPC
10
Trigonometry: Special angles.
Name:______Key______
(0°, 30°, 45°, 60°, 90°)
Equilateral Triangles.
3 equilateral triangles with different side lengths are given below.
All equilateral triangles have 3 equal sides and 3 interior angles of 60°.
If we draw a line labeled as height from the vertex to the base line in each
equilateral triangle, then we have 2 identical right triangles in each diagram as
shown below. These right triangles all have special interior angles of 30°-60°-90°.
tan 60° =
3  1.73205 ,
tan 60° =
2 3
 3,
2
25 3
25
= 3
tan 60° =
3 is the exact value.
sin 60° =
3
2
,
sin 60° =
cos 30° =
3
2
,
cos 30° =
cos 60° =
1
2
,
3
2 3

,
2
4
cos 60° =
3
2 3

,
2
4
2 1
 ,
4 2
Did you notice that sin 60° = cos 30°?
sin 60° =
25 3
3

50
2
cos 30° =
25 3
3

50
2
cos 60° =
25 1

50 2
Isosceles Right Triangle.
3 isosceles right triangles with different side lengths are given below.
These isosceles right triangles have 2 equal sides and 2 interior angles of 45°.
These right triangles all have special interior angles of 45°-90°-45°.
tan 45° =
1
1
1
sin 45° =
1
2
,
sin 45° =
cos 45° =
1
2
,
cos 45° =
,
tan 45° =
2
1
2
2
2 2
2
2 2
,


1
,
2
1
,
2
tan 45° =
13
1
13
sin 45° =
cos 45° =
13
1

13 2
2
13
1

13 2
2