Question

Given the value of cos 55° ≈ 0.5736, enter the sine of a complementary angle. Use an expression relating trigonometric ratios of complementary angles.

Answer 1

Answer: $\cos(55^{\circ}) = \boxed{\sin(35^{\circ})} \approx \boxed{0.5736}$

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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.