Trigonometric Ratios

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Step-by-step guide to trigonometric ratios

Trigonometric ratios are the ratio of the lengths of the sides of a triangle. These ratios in trigonometry relate the ratio of the sides of a right triangle to the corresponding angle. There are six trigonometric ratios, namely, sine, cosine, tangent, secant, cosecant, and cotangent. These ratios are written as \(sin\), \(cos\), \(tan\), \(sec\), \(cosec\) (or \(csc\)), and \(cot\) in short.

The values of these trigonometric ratios can be calculated using the measurement of an acute angle, \(θ\), in a right triangle.

What are trigonometric ratios?

These six trigonometric ratios can be defined as:

Sine: The sine ratio for any given angle is defined as the ratio of the perpendicular to the hypotenuse. In the given triangle, the \(sin\) of the angle \(θ\) can be considered as follows, \(\color{blue}{sin\: θ = \frac{AB}{AC}}\).

Cosine: The cosine ratio for any given angle is defined as the ratio of the base to the hypotenuse. In the given triangle, the \(cos\) of the angle \(θ\) can be considered as follows, \(\color{blue}{cos\: θ = \frac{BC}{AC}}\).

Tangent: The tangent ratio for any given angle is defined as the ratio of the perpendicular to the base. In the given triangle, the \(tan\) of the angle \(θ\) can be considered as follows, \(\color{blue}{tan\: θ = \frac{AB}{BC}}\).

Cosecant: The cosecant ratio for any given angle is defined as the ratio of the hypotenuse to the perpendicular. In the given triangle, the \(cosec\) of the angle \(θ\) can be considered as follows, \(\color{blue}{cosec\: θ = \frac{AC}{AB}}\).

Secant: The secant ratio for any given angle is defined as the ratio of the hypotenuse to the base. In the given triangle, the \(sec\) of the angle \(θ\) can be considered as follows, \(\color{blue}{sec\: θ = \frac{AC}{BC}}\).

Cotangent: The cotangent ratio for any given angle is defined as the ratio of the base to the perpendicular. In the given triangle, the \(cot\) of the angle \(θ\) can be considered as follows, \(\color{blue}{cot\: θ = \frac{BC}{AB}}\).

Trigonometric ratios formulas

We can use the shorthand form of trigonometric ratios to compare the length of both sides with the base angle. The angle \(θ\) is acute \((θ<90º)\) and in general is measured with reference to the positive \(x\)-axis, in the anticlockwise direction. The basic trigonometric ratio formulas are given below,

• \(\color{blue}{sin\: θ = \frac{Perpendicular}{Hypotenuse}}\)
• \(\color{blue}{cos\: θ = \frac{Base}{Hypotenuse}}\)
• \(\color{blue}{tan\: θ = \frac{Perpendicular}{Base}}\)
• \(\color{blue}{sec\: θ =\frac{Hypotenuse}{Base}}\)
• \(\color{blue}{cosec\: θ = \frac{Hypotenuse}{Perpendicular}}\)
• \(\color{blue}{cot\: θ = \frac{Base}{Perpendicular}}\)

The new set of formulas for trigonometric ratios is:

• \(\color{blue}{sin\: θ = \frac{1}{cosec\: θ}}\)
• \(\color{blue}{cos\: θ = \frac{1}{sec\: θ}}\)
• \(\color{blue}{tan\: θ = \frac{1}{cot\: θ}}\)
• \(\color{blue}{cosec\: θ = \frac{1}{sin\: θ}}\)
• \(\color{blue}{sec\: θ = \frac{1}{cos\: θ}}\)
• \(\color{blue}{cot\: θ = \frac{1}{tan\: θ}}\)

Trigonometric ratios table

The value of trigonometric ratios for specific angles is in the table below:

Trigonometric Ratios – Example 1:

Find the value of \(tan\:θ\) if \(sin\:θ\:=\frac{10}{3}\) and \(cos\:θ\:=\frac{5}{3}\:\).

Solution: Use the formula of the trigonometric ratio to solve this problem: \(tan\: θ = \frac{Perpendicular}{Base}\).

\(tan\:θ =\frac {10}{5}=2\)