relation between arc, radius and central angle. Proof of the trigonometrical ratios of complementary angles

Relation between arc radius and angle

Arc length   =   circumference • [central angle (degrees) ÷ 360]

Proof of the trigonometric ratios of complementary allied angles

Two acute angles are complementary to each other if their sum is equal to 90°. In a right triangle the sum of the two acute angles is equal to 90°. So, the two acute angles of a right triangle are always complementary to each other.

Let ABC be a right triangle, right angled at B

If <ACB = θ, then <BAC = 90° – θ and hence the angles <BAC and <ACB are complementary

For the angle θ, we have

Similarly, for the angle (90° – θ), we have

Comparing the equations in (1) and (2) we get,

Trigonometric Ratios of Complementary Angles

Examples: Evaluate  :  cos 56° / sin 34°

The angles 56° and 34° are complementary.

So, using trigonometric ratios of complementary angles, we have

cos 56°  =  cos (90° – 56°)  =  sin 34°

cos 56° / sin 34°  =  sin 34° / sin 34°  =  1

Hence the value of cos 56° / sin 34° is 1.

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