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The 6 trigonometric ratios can be said to be the sine (sin), the cosine (cos), the tangent (tan), the cotangent (cot), the cosecant (cosec), and the secant (sec). In the geometry, the trigonometry can be said to be a branch of the mathematics that actually deals with the sides as well as the angles of any right-angled triangle. Hence, the trig ratios are actually evaluated in relation to the sides as well as the angles.

The trigonometry ratios in relation to a specific angle ‘θ’ has been given below:

Trigonometric Ratios | |

Sin θ | Opposite Side to the θ/Hypotenuse |

Cos θ | Adjacent Side to the θ/Hypotenuse |

Tan θ | Opposite Side/Adjacent Side and Sin θ/Cos θ |

Cot θ | Adjacent Side/Opposite Side and 1/tan θ |

Sec θ | Hypotenuse/Adjacent Side and 1/cos θ |

Cosec θ | Hypotenuse/Opposite Side and 1/sin θ |

Trigonometric Ratios can be said to be defined as the values of every trigonometric function is actually based upon the value in relation to the ratio of the sides in any right-angled triangle.* *The ratios of the sides of any right-angled triangle in relation to any of its critical or acute angles are known to be the trigonometric ratios of that specific angle.

Trigonometric ratios are possible to be calculated by taking the specific ratio of any 2 sides of the right-angled triangle. One shall be able to evaluate the 3^{rd} side using the Pythagoras theorem, provided the measure of the other 2 sides. One shall be able to utilize the abbreviated version of the trigonometric ratios in order to compare the length of any 2 sides with the specific angle in the particular base. The angle θ can be said to be an acute angle (θ < 90º) and in general has been measured with reference to positive x-axis, in the anti-clockwise direction. The rudimentary trigonometric ratios formulas have been given below: -

- sin θ = the Perpendicular/the Hypotenuse
- cos θ = the Base/the Hypotenuse
- tan θ = the Perpendicular/the Base
- sec θ = the Hypotenuse/the Base
- cosec θ = the Hypotenuse/the Perpendicular
- cot θ = the Base/the Perpendicular

Therefore, one should first observe the reciprocal trigonometric ratio formulas in relation to the above-mentioned trigonometric ratios. As one actually observes, he or she notices that sin θ is actually a reciprocal of the cosec θ, the cos θ would be a reciprocal of the sec θ, the tan θ can be said to be a reciprocal of the cot θ, as well as vice-versa. Hence, the new set or series of the formulas for the trigonometric ratios is: -

- the sin θ = 1/cosec θ
- the cos θ = 1/sec θ
- the tan θ = 1/cot θ
- the cosec θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ

In the particular trigonometric ratios table, one actually utilizes the values of the trigonometric ratios for the purposes of the standard angles 0°, 30°, 45°, 60°, as well as 90º. It can be said to be very easy to actually predict the values in relation to the table as well as to utilize the table as a locus in order to calculate the values in relation to the trigonometric ratios for the numerous other angles, through the utilization of the trigonometric ratio formulas for the existing patterns in the trigonometric ratios as well as amidst the angles. After such process, one shall be able to summarize the value in relation to the trigonometric ratios for the particular angles in the specific table that has been mentioned below: -

Apart from the astronomy as well as the geography, the trigonometry can be said to be applicable in numerous fields such as the satellite navigation, developing the computer music, the chemistry number theory, the medical imaging, the electronics, the electrical engineering, the civil engineering, the architecture, the mechanical engineering, the oceanography, the seismology, the phonetics, the image compression as well as the game development. It should be noted that there are Trigonometry Applications in the real life. It might not have the direct applications in solving the practical issues but has been utilized in numerous fields. For instance, the trigonometry is actually utilized in developing the computer music: as one would be familiar that sound actually travels in the version of the waves and such wave pattern, with the help of a sine or a cosine function for the development of the computer music. Following can be said to be few applications where the trigonometry as well as its functions can be said to be applicable: -

- Trigonometry to Measure Height of a Building or a Mountain
- Trigonometry in Aviation
- Trigonometry in Criminology
- Trigonometry in Marine Biology
- Trigonometry in Navigation

Example: If the distance from where the building is observed is 90 ft from its base and the angle of elevation to the top of the building is 35°, then find the height of the building.

Solution: Given:

- Distance from the building is 90 feet from its base.
- The angle of elevation from to the top of the building is 35°.

Now, one should search for the height of the building by recalling the trigonometric formulas. Here, the angle as well as the adjacent side length are actually provided. Therefore, through the utilization of the formula of the tan.

tan35∘=Opposite Side/Adjacent Side

tan 35°= h/90

h = 90 × tan 35°

h = 90 × 0.7002

h = 63.018 ft

Therefore, the height of the specific building is actually 63.018 ft.

The 3 primary trig ratios can be said to be the sine (sin), the cosine (cos) and the tangent (tan).

SOH CAH TOA can be said to be a mnemonic device that is actually utilized to remember the particular ratios of the sine, the cosine, and the tangent in the trigonometry.

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