Trigonometric ratios - Maths & Stats - Trigonometry - The Library at Leeds Beckett University

Learning Outcomes

If you work through this section you should be able to:

Understand the key definition of trigonometry.
Understand the trigonometric ratios.
Find sin, cos and tan of angles.
Calculate cosec, sec and cot of angles.

Trigonometric ratios
Reciprocal ratios

The trigonometric ratios are defined as:

sin $θ = \frac{o p p o s i t e}{h y p o t e n u s e} = \frac{A B}{B C}$

cos $θ = \frac{a d j a c e n t}{h y p o t e n u s e} = \frac{A C}{B C}$

tan $θ = \frac{o p p o s i t e}{a d j a c e n t} = \frac{A B}{A C}$

Example

Trigonometric ratio example.

Using the above triangle,

sin $θ = \frac{o p p o s i t e}{h y p o t e n u s e} = \frac{3}{5} = 0.6$

cos $θ = \frac{a d j a c e n t}{h y p o t e n u s e} = \frac{4}{5} = 0.8$

tan $θ = \frac{o p p o s i t e}{a d j a c e n t} = \frac{3}{4} = 0.75$

In addition to the three trigonometric ratios there are three reciprocal ratios, namely the cosecant (csc), the secant (sec) and the cotangent (cot). The reciprocal ratios are defined as:

csc $θ = \frac{1}{sin θ} = \frac{h y p o t e n u s e}{o p p o s i t e}$

sec $θ = \frac{1}{cos θ} = \frac{h y p o t e n u s e}{a d j a c e n t}$

cot $θ = \frac{1}{tan θ} = \frac{a d j a c e n t}{o p p o s i t e}$

Examples

Reciprocal trigonometric ratios exercise diagram

Using the above triangle,

csc $θ = \frac{1}{sin θ} = \frac{h y p o t e n u s e}{o p p o s i t e} = \frac{5}{3} = 1.67$

sec $θ = \frac{1}{cos θ} = \frac{h y p o t e n u s e}{a d j a c e n t} = \frac{5}{4} = 1.25$

cot $θ = \frac{1}{tan θ} = \frac{a d j a c e n t}{o p p o s i t e} = \frac{4}{3} = 1.33$

Activity

You can download a version of this Trigonometric ratios activity in Word format:

Trigonometric ratios activity

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