Co-Functions

Chapter 1.4

A Worthwile Discovery

Did you notice while completing exercises related to the previous section, that sin 10⁰ was the same as cos 80⁰? Sin 25⁰ = cos 65⁰. Also cos 58⁰ = sin 32⁰ and sin 58⁰ = cos 32⁰.

Complementary Angles

10⁰ and 80⁰ are complementary angles. So are 25⁰ with 65⁰ and 58⁰ with 32⁰.

We shall, therefore, say that when we have complementary angles, the sine of one angle is always equal to the cosine of the other angle. Does this statement explain why sin 45⁰ equals cos 45⁰? Prove sin 45⁰ equals cos 45⁰ in another way.

Theorem

When two angles are complementary, the sine of one angle equals the cosine of the other.

Additional Ratios

In addition to the sine, cosine, and tangent of an angle, there are three other useful ratios.

Cotangent

The cotangent of an angle is the ratio of the adjacent leg to the opposite leg. For example, from the diagram above:

        adjacent leg   a
cot B = ------------ = -
        opposite leg   b

What is cot A?

Secant

The secant (abbreviated sec) of an angle is the ratio of the hypotenuse to the adjacent leg. For example, from the diagram above:

        hypotenuse     c
sec B = ------------ = -
        adjacent leg   a

What is sec A?

Cosecant

The cosecant (abbreviated csc) of an angle is the ratio of the hypotenuse to the opposite leg. For example, from the diagram above:

        hypotenuse     c
csc B = ------------ = -
        opposite leg   b

What is csc A?

Theorem

When two angles are complementary, the tangent of one angle equals the cotangent of the other; the secant of one angle equals the cosecant of the other.

NOTE - These theorems explain why it's not necessary to work out functions for angles between 45⁰ and 90⁰ if the functions between 0⁰ and 45⁰ have already been obtained.

Pages 9-11

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