17.3 CHAPTER 17. TRIGONOMETRY
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45 ◦
Figure 17.7: An isosceles right angled triangle.Take any right-angled triangle with one angle 45 ◦. Then, because one angle is 90 ◦, the third angle is
also 45 ◦. So we have an isosceles right-angled triangle asshown in Figure 17.7.
If the two equal sides are of length a, then the hypotenuse, h, can be calculated as:
h^2 = a^2 + a^2
= 2a^2
∴ h =√
2 aSo, we have:
sin(45◦) =
opposite(45◦)
hypotenuse
=
a
√
2 a=1
√
2
cos(45◦) =adjacent(45◦)
hypotenuse
=
a
√
2 a
=1
√
2
tan(45◦) =
opposite(45◦)
adjacent(45◦)
=
a
a
= 1We can try something similar for 30 ◦and 60 ◦. We start with an equilateral triangle and we bisect one
angle as shown in Figure 17.8. This gives us theright-angled triangle that we need, with one angle of
30 ◦and one angle of 60 ◦.
If the equal sides are oflength a, then the base is^12 a and the length of the vertical side, v, can be
calculated as:
v^2 = a^2 − (1
2
a)^2= a^2 −1
4
a^2=3
4
a^2∴ v =√
3
2
a