Till now you must be familiar with the Pythagorean Theorem but what if you are not given all sides? What if you have an angle and just one side? You can't apply the Pythagorean theorem because it requires ATLEAST two sides to find the third side. That is where trigonometric ratios came in. Trigonometric Ratios are the trigonometric functions based on the value of the ratio of sides. These ratios are applied on a right angle triangle. There are three common trigonometric ratios and they are \sin {} \quad \cos {} \quad \tan {}. However, these are not all trigonometric ratios, there are more trigonometric ratios and we will discuss them too in this lesson.

To apply these ratios, you need to understand every element of a right-angle triangle. A right-angle triangle looks like this:

The best way to identify whether the triangle is a right angle triangle or not is to check the angles of the triangle. If one of the angles is equal to 90 degrees that means the triangle is a right-angle triangle. Once you identified that you are working with a right-angle triangle, now let's break the triangle into its sides. There are three different sides of a right-angled triangle and they are:

  • Hypotenuse (the longest side and represented by "a")
  • Perpendicular (opposite side to the angle)
  • Base (Adjacent side to the angle)

In a right triangle, the following trigonometric ratios can be defined:

Sine

The sine of the angle B is the ratio between the length of the opposite side and the hypotenuse of the triangle.

It is denoted by \sin { B }.

\sin { B } = \frac { opposite side/perpendicular }{ hypotenuse } = \frac { b }{ a }

Cosine

The cosine of the angle B is the ratio between the length of the adjacent side and the hypotenuse of the triangle.

It is denoted by \cos { B }.

\cos { B } = \frac { adjacent side/base }{ hypotenuse } = \frac { c }{ a }

 

Tangent

The tangent of angle B is the ratio between the length of the opposite side and the adjacent side of the triangle.

It is denoted by \tan { B }.

\tan { B } = \frac { opposite side/perpendicular }{ adjacent side/base } = \frac { \sin { B } }{ \cos { B } } = \frac { b }{ c }

 

Cosecant

The cosecant of angle B is the inverse of the sine of B.

It is denoted by \csc { B }.

\csc { B } = \frac { 1 }{ \sin { B }} = \frac { hypotenuse }{ opposite side/perpendicular } = \frac { a }{ b }

 

Secant

The secant of angle B is the inverse of the cosine of B.

It is denoted by \sec { B }.

\sec { B } = \frac { 1 }{ \cos { B }} = \frac { hypotenuse }{ adjacent side/base } = \frac { a }{ c }

 

Cotangent

The cotangent of angle B is the inverse of the tangent of B.

It is denoted by \cot { B }.

\cot { B } = \frac { 1 }{ \tan { B }} = \frac { \cos { B } }{ \sin { B }} = \frac { adjacent side/base }{ opposite side/perpendicular } = \frac { c }{ b }

Easy Trick to Remember all Trigonometric Ratios

The first three trigonometric ratios are the most important ratios. Once you memorize them, memorizing as well as understanding other ratios will be very easy for you. Hence, you need to memorize these trigonometric ratios:

  • \sin {  }
  • \cos {  }
  • \tan {  }

We can write these formula using the first alphabet of the word, for example:

\sin { x } = \frac { P }{ H } \qquad \cos { x } = \frac { B }{ H } \qquad \tan { x } = \frac { P }{ B }

Where,

stands for Perpendicular,

H stands for Hypotenuse,

and stands for Base.

Here is a small trick to remember, you need to memorize this line, "Some People Have Curly Brown Hair Through Proper Brushing". Let's break it into three parts, "Some People Have", "Curly Brown Hair", and "Through Proper Brushing". If you notice the first alphabet of these sentences then you might notice that they match with the trigonometric ratios. For example, in the first part, the "S" in some stands for \sin {} and "P" and "H" in the "People Have" stands for Perpendicular and Hypotenuse. In the same way, if you check the first alphabet of the second part (which is "Curly Brown Hair"), the "C" in "Curly" stands for \cos {} and "B" and "H" in the "Brown Hair" stands for Base and Hypotenuse. Last but not least, the "T" in "Through" stands for \tan {}, however, the "P" and "B" in the "Proper Brushing" stands for Perpendicular and Base.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.