The equalities that involve trigonometric functions and hold true for every value involved, such that both sides of the equalities are justified, are defined as trigonometric identities. In this article, we will try to learn and explore trigonometric identities. There are three most significant trigonometric ratios, sine, cosine, and tangent. The other three trigonometric ratios are derived respectively by the above-mentioned ratios i.e., secant, cosecant, and cotangent. Can you guess whether all the ratios are connected to each other or not? No? Let me tell you all these trigonometric ratios are connected to each other through trigonometric identities. We will try to understand the trigonometric identities in the following sections below. But before that let us see how a trigonometric ratios table looks like.
Trigonometric Ratios Table
The trigonometric table helps us to understand and grasp the concept of trigonometry in an efficient manner. The values of various trigonometric ratios for standard angles such as 0°, 30°, 45°, 60°, and 90° present in a tabular form is known as the trigonometric ratio table. The trigonometric ratios table comprises trigonometric ratios such as sine, cosine, and tangent. The other three are cotangent, secant, and cosecant. These values are very important in order to calculate various problems regarding trigonometry. The values of complementary angles such as 30° and 60° can be computed using trigonometric formulas for various trigonometric ratios. Let us now see how the trigonometric ratios look like.
| Radians | Degrees | Sin θ | Cos θ | Tan θ | Cosec θ | Sec θ | Cot θ |
| 0 | 0° | 0 | 1 | 0 | Not defined | 1 | Not defined |
| π/6
|
30° | 1/2 | √3/2 | 3√3 | 2 | 2√3/3 | √3 |
| π/4
|
45° | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| π/3 | 60° | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| π/2 | 90° | 1 | 0 | Not defined | 1 | Not defined | 0 |
Some Important Trigonometric Identities
There are various types of trigonometric identities, but there are very few which are very significant and that comes across in mathematical problems. Let us try to understand some important trigonometric identities. You can also learn it easily through cuemath. It’s an online Maths tutoring platform where you can learn Maths concepts easily in an interesting way.
- Reciprocal Trigonometric Identities:
As mentioned above, the reciprocal of sine, cosine, and tan are sec, cosec, and cot respectively. Therefore, the reciprocal trigonometric identities are as follows:
- Sin θ = 1/Cosec θ or cosec θ = 1/sin θ
- Cos θ = 1/sec θ or sec θ = 1/cos θ
- Tan θ = 1/cot θ or cot θ = 1/tan θ
- Pythagorean Trigonometric Identities:
The Pythagorean theorem basically derives the Pythagorean trigonometric identities. If we apply the Pythagorean identities in a right-angled triangle, we get,
Opposite square + Adjacent square = Hypotenuse square.
If we divide both sides by hypotenuse squares. Then,
- Opposite square/ hypotenuse square + adjacent square /hypotenuse square = hypotenuse square/hypotenuse square.
- Sine θ = Cos.Cos θ.
This is one of the Pythagorean trigonometric identities. By this identity, we can derive two other trigonometric identities.
- 1 + tan.tan θ = sec.sec θ
- 1 + cot.cot θ = cosec.cosec θ
- Complementary and Supplementary trigonometric identities:
The pair of two angles such that their sum is equal to 90 degrees is known as the complementary angle. The trigonometric identities of complementary angles are as follows:
- Sin (90° -θ) = cos θ
- Cos (90°-θ) = sin θ
- Cosec (90°-θ) = sec θ
- Sec (90°– θ) = cosec θ
- Tan (90°-θ) = cot θ
- Cot (90-θ) = tan θ
The pair of two angles such that their sum is equal to 180 degrees is known as the supplementary angle. The trigonometric identities of supplementary angles are as follows.
- Sin (180° – θ) = sin θ
- Cos (180° – θ) = -cos θ
- Cosec (180° – θ) = cosec θ
- Sec (180°- θ) = -sec θ
- Cot (180° -θ) = – cot θ
- Tan (180° -θ) = tan θ
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