Trigonometric Ratios

What are trigonometric ratios?

Sine: In the given triangle, the \(sin\) of the angle \(θ\) can be considered as follows, \(\color{blue}{sin\: θ = \frac{AB}{AC}}\).

Cosine: In the given triangle, the \(cos\) of the angle \(θ\) can be considered as follows, \(\color{blue}{cos\: θ = \frac{BC}{AC}}\).

Tangent: In the given triangle, the \(tan\) of the angle \(θ\) can be considered as follows, \(\color{blue}{tan\: θ = \frac{AB}{BC}}\).

Cosecant: In the given triangle, the \(cosec\) of the angle \(θ\) can be considered as follows, \(\color{blue}{cosec\: θ = \frac{AC}{AB}}\).

Secant: In the given triangle, the \(sec\) of the angle \(θ\) can be considered as follows, \(\color{blue}{sec\: θ = \frac{AC}{BC}}\).

Cotangent: 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,

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

Trigonometric Ratios – Example 1:

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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: \(tan\: θ = \frac{Perpendicular}{Base}\).

\(tan\:θ =\frac {10}{5}=2\)

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