Solving Trigonometry: 4.2 Trigonometric Ratios and Special Angles

Trigonometric Ratios

The relationships between the angles and the sides of a right triangle are expressed in terms of TRIGONOMETRIC RATIOS.

Figure 1. A right angle triangle

A right triangle has one of its interior angles measuring 90° (a right angle).
Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.

In a right triangle, the six trigonometric ratios are;

sine
cosine
tangent
cosecant
secant
cotangent

Figure 2. Hypotenuse, Opposite and Adjacent

Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$

Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$

Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.$

Cosecant csc(A), or cosec(A), is the reciprocal of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:

$\csc A = \frac {1}{\sin A} = \frac {\textrm{hypotenuse}} {\textrm{opposite}} = \frac {h} {a}.$

Secant sec(A) is the reciprocal of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:

$\sec A = \frac {1}{\cos A} = \frac {\textrm{hypotenuse}} {\textrm{adjacent}} = \frac {h} {b}.$

Cotangent cot(A) is the reciprocal of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:

$\cot A = \frac {1}{\tan A} = \frac {\textrm{adjacent}} {\textrm{opposite}} = \frac {b} {a}.$

Special Angles

The values of trig functions of specific angles can be represented by known ratios, and are good to commit to memory to help you work through problems faster.

These 'special' angles can be remembered by examining two different triangles.

Specifically, the trig functions for angles of 0, 30, 45, 60, 90 degrees are the special ones.