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Answered by
0
take the complex exponential defined as its series and consider the complex definitions of the trigonometric functions:
cos()=
+
−
2
∧sin()=
−
−
2
,
for all ∈ℂ.
cos
(
z
)
=
e
i
z
+
e
−
i
z
2
∧
sin
(
z
)
=
e
i
z
−
e
−
i
z
2
i
,
for all
z
∈
C
.
Take
∈ℝ
θ
∈
R
. The following holds:
(cos()
)
2
+(sin()
)
2
=
2
+2+
−2
4
−
2
−2+
−2
4
=
2−(−2)
4
=1.
Answered by
1
Consider a triangle ABC, in which AC is the longest side, and angle ABC = 90°, So:
AC = Hypotenuse
AB = Perpendicular
BC = Base
By Pythagoras theorem:
AC² = AB² + AC² _______(i)
Now consider, ø = angle ACB
sinø = P/H = AB/AC
cosø = B/H = BC/AC
Now, sin²ø = AB²/AC²
and, cos ø = BC²/AC²
Add both,
son²ø + cos²ø = AB²/AC² + BC²/AC²
=(AB² + BC²)/AC²
from equation (i):
= (AC²)/AC² = 1
So, son²ø + cos²ø = 1
Hence proved.
Thankyou!!!
AC = Hypotenuse
AB = Perpendicular
BC = Base
By Pythagoras theorem:
AC² = AB² + AC² _______(i)
Now consider, ø = angle ACB
sinø = P/H = AB/AC
cosø = B/H = BC/AC
Now, sin²ø = AB²/AC²
and, cos ø = BC²/AC²
Add both,
son²ø + cos²ø = AB²/AC² + BC²/AC²
=(AB² + BC²)/AC²
from equation (i):
= (AC²)/AC² = 1
So, son²ø + cos²ø = 1
Hence proved.
Thankyou!!!
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