Answer: [tex]\cos(55^{\circ}) = \boxed{\sin(35^{\circ})} \approx \boxed{0.5736}[/tex]
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Explanation:
The rule to use is
cos(x) = sin(y) if and only if x+y = 90
Proof of this statement is shown in the section below.
We say that angles x and y are complementary angles since they add to 90 degrees.
Use of a calculator shows,
cos(55) = 0.573576
sin(35) = 0.573576
both values being approximate.
note how 55+35 = 90.
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Proof of the claim that if x+y = 90, then cos(x) = sin(y)
Let x and y be the acute angles of a right triangle. See the diagram below. Sides a,b,c are such that 'a' is opposite angle x, b is opposite angle y, and c is the hypotenuse.
We can then say,
cos(angle) = adjacent/hypotenuse
cos(x) = b/c
and also,
sin(angle) = opposite/hypotenuse
sin(y) = b/c
Both are equal to b/c, which means cos(x) = sin(y)
x+y = 90 is true for any pair of acute angles of a right triangle.
