# Trigonometric Ratios

Trigonometry is a branch of mathematics that deals with the relationship between the angles and sides of a right triangle. This step-by-step guide teaches you trigonometric ratios. ## Step-by-step guide totrigonometric 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: $$tan\: θ = \frac{Perpendicular}{Base}$$.

$$tan\:θ =\frac {10}{5}=2$$

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