There are six Trig Identities in total, we use them to solve all the trigonometric problems. All these trig identities rely on the trigonometric ratio such as Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant, etc. So, how do we describe these ratios, and do we use an object to represent them? Ta-da! We use a triangle to represent these ratios. A triangle has three sides, where the base is “B”, height is perpendicular “P”, and the longest side is Hypotenuse “H”. Now, let us begin our journey with the story of Trigonometric Ratios moving across the boundary of a triangle.
Story of Trigonometric Identities Moving across the Boundary of a Triangle
- For Sin , ”P” is the first floor of the building, while “H” is the ground floor., so here the story goes:
Saina (Sin ) lives on the first floor and she is very hungry. On the ground floor, there is a confectionery shop and to reach there, she has to come down to the ground floor, so, she goes from P to H, which we represent as;
Sin = P/H …..(1)
- For Cosec :
However, Cosec takes an opposite path, i.e., from H to P, for this we have:
Cosec = H/P
Or, Sin = 1/Cosec ….(2)
- For Cos :
Your mother doesn’t want you to give it up because (Cos ) she wants you to be “B” successful and stay happy “H” always, so we represent this positive quote as;
Cos = B/H
- For Sec :
Now, if you get happy “H” after succeeding, always remember that battleground “B” of life is endless, so now we will represent this phrase for Sec as;
Sec = H/B
- For Tan
We often see that people after gaining a lot of power “P” leave behind their beloved ones. Do you think it is right? Well, it is not. So, now we will represent this sentence for Tan as;
Tan = P/B
- Cot
A person with gratefulness remembers their bad days “B” even after they gain a higher position “P” in their lives. So, now we will represent this for Cot as;
Cot = B/P
The above identities that we discussed above are Reciprocal Trigonometric Identities.
Trigonometric Ratio Table
The trigonometric ratio has the following formulas:
- Tan = Sin /Cos
- Cot = Cos /Sin
Now, we will replace “” with values starting from 0 to 90 and we get the following values under the following Trigonometric Ratio Table:
| 0° | 30° | 45° | 60° | 90° | |
| Sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| Cosec | ∞ | 2 | √2 | 2/√3 | 1 |
| Tan | 0 | 1/√3 | 1 | √3 | ∞ |
| Cot | ∞ | √3 | 1 | 1/√3 | 0 |
| Cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| Sec | 1 | 2/√3 | √2 | 2 | ∞ |
Here, ∞ means not defined or 1/0
How do Trigonometric Identities get affected by Opposite Angles?
- Sin (- ) = – Sin
- Cos (- ) = Cos
- Tan (- ) = – Tan
- Cot (- ) = – Cot
- Sec (- ) = Sec
- Cosec (- ) = – Cosec
Here, we see that Cos is very powerful. Though the negative sign (negative of the angle) attacked over it, it remained positive, while others got negative.
Add Sugar To Coffee Trigonometry
Using Add Sugar To Coffee, we will find Trigonometric Identities get affected by Complementary Angles, as shown below:
| Angle | Positive | Negative | |
| Add (All) | 0° – 90° | Sin , Cos , Tan , Cot , Sec , and Cosec | None |
| Sugar | 90° – 180° | Sin , Cosec | Cos , Tan , Cot , Sec |
| To | 180° – 270° | Tan , Cot | Sin , Cosec , Cos , Sec |
| Coffee | 270° – 360° | Cos , Sec | Sin , Cosec , Tan ,Cot |
How do Trig Identities get affected by Complementary Angles?
Suppose Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant used “” of energy out of its total energy of 90° for work out. Now, how much energy do they have to run 2 km ahead? So, here, the left out energy is the complementary angle, i.e., 90° – . Now, from the above table, let us find it out complementary angles:
- Sin (90° – ) = Cos
- Cos (90° – ) = Sin
- Cosec (90° – ) = Sec
- Sec (90° – ) = Cosec
- Tan (90° – ) = Cot
- Cot (90° – ) = Tan
How do Trigonometric Identities get affected by Supplementary Angles?
Suppose Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant used “” of energy out of its total energy of 180° for completing their homework. Now, how much energy do they have to play cricket? So, here, the left out energy is the supplementary angle, i.e., 180° – . Now, from the above table, let us find it out complementary angles:
- Sin (180° – ) = Sin
- Cos (180° – ) = – Cos
- Cosec (180° – ) = Cosec
- Sec (180° – ) = – Sec
- Tan (180° – ) = – Tan
- Cot (180° – ) = – Cot
This was all about Trigonometric Identities and Ratios with story-telling methods to help you remember these identities forever. These can be learnt in an engaging way from Cuemath.