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The ratios of the lengths of the sides of a triangle are known as trigonometric ratios. In trigonometry, these ratios link the ratio of sides of the right triangle to the angle. The sin, cos, and tan can be used to calculate the other key trigonometric ratios, cosec, sec, and cot.
The name “trigonometry” comes from the Greek words “trigonon” and “metron,” which both mean “to measure.” It’s a branch of mathematics with the relationship between a right-angled triangle’s angles and sides. Trigonometry is, in reality, one of the oldest sciences studied by experts all over the globe.
Trigonometric Ratios: What Are They?
The 6 trigonometric ratios in trigonometry are Sine, cosine, tangent, secant, cosecant, and cotangent. In summary, these ratios are sin, cos, tan, sec, cosec (or csc), and cot. Trigonometric ratios could be used to calculate the ratios of any two of a right-angled triangle’s three sides in terms of their respective angles.
In any right-angled triangle ABC, the values of these trigonometric ratios can be computed using the measure of an acute angle. This means that the value of a ratio of any two triangle sides is determined by angle C. In that situation, only the base and perpendicular will swap for the specified right triangle.
TRIGONOMETRIC RATIOS FORMULAS
The ratio of any 2 sides of a right-angled triangle can be used to derive trigonometric ratios. Given the measurement of the other two sides, we may use the Pythagoras theorem to calculate the third side. To compare the length of any 2 sides with the angle in the base, we can utilise the abbreviated form of trigonometric ratios. The angle is sharp (90 ̊) and is generally measured in the anticlockwise direction with reference to the positive x-axis. The formulas for basic trigonometric ratios are listed here.
sinϴ = perpendicular/hypotenuse
cosϴ = base/hypotenuse
tanϴ = perpendicular/base
secϴ = hypotenuse/base
cosecϴ = hypotenuse/perpendicular
cotϴ = base/perpendicular
Let’s look at the reciprocal trigonometry ratio formulas for the trigonometric ratios described previously. We can see that sin is a reciprocal of cosec, cos is a reciprocal of sec, tan is a reciprocal of cot, and vice versa as we see. As a result, the new set of trigonometric ratio formulas is as follows:
sinϴ = 1/cosecϴ
cosϴ = 1/secϴ
tanϴ = 1/cotϴ
secϴ = 1/cosϴ
cosecϴ = 1/sinϴ
cotϴ = 1/tanϴ
TRIGONOMETRY TABLE
The values of trigonometric standard angles like 0°, 30°, 45°, 60°, and 90° can be found using the trigonometric ratios table. The trigonometric ratios are defined as sine, cosine, tangent, cosecant, secant, and cotangent. Sin, cos, tan, cosec, sec, and cot are abbreviated forms of these ratios.
To answer trigonometry issues, the value of trigonometric ratios of standard angles are required. As a result, the value of the trigonometric ratios of such standard angles must be remembered.
Refer to the table for typical angles that can be utilized to solve numerous trigonometric ratio problems.
Angles | 0 ̊ | 30 ̊ | 45 ̊ | 60 ̊ | 90 ̊ |
Sin ϴ | 0 | ½ | 1/√2 | √3/2 | 1 |
Cos ϴ | 1 | √3/2 | 1/√2 | ½ | 0 |
Tan ϴ | 0 | 1/√3 | 1 | √3 | ∞ |
Cosec ϴ | ∞ | 2 | √2 | 2/√3 | 1 |
Sec ϴ | 1 | 2/√3 | √2 | 2 | ∞ |
Cot ϴ | ∞ | √3 | 1 | 2/3 | 0 |
Similarly, we may determine the trigonometric ratio values for angles other than 90 degrees, such as 180 degrees, 270 degrees, and 360 degrees.
TRIGONOMETRIC IDENTITIES
The identities are geometrically related to specific trigonometric functions (like sine, cosine, and tangent) of one or even more angles.
The major trigonometry functions are sine, cosine, and tangent, while the other 3 functions are cotangent, secant, and cosecant.
Only the right-angle triangle has the trigonometric identities. The sides of the right triangle, like the adjacent, opposing, and hypotenuse sides, are used to define all of these trigonometric ratios.
Complementary Angles’ Trigonometric Identities
Two angles are said to be complementary in geometry if their sum equals 90 degrees. Similarly, the trigonometric identities for complementary angles can be learned here.
Sin (90° – ϴ) = Cos ϴ
cos (90° – ϴ) = sin ϴ
Tan (90° – ϴ) = Cot ϴ
Cot (90° – ϴ) = Tan ϴ
Sec (90° – ϴ) = csc ϴ
Csc (90° – ϴ) = Sec ϴ
CONCLUSION
If the angle keeps the same, the values of trigonometric ratios do not change when the side lengths of the triangle change. The missing sides and angles in triangle are found using trigonometric ratios.
Trigonometry deals with the sides and angles of a right-angled triangle. The heart of trigonometry issues is that we must find the remaining sides and angles of a triangle when some variables are provided.
This can be accomplished by using a proper ratio of a triangle’s side to its acute angle. Trigonometric ratios of angles are the ratios of acute angles. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec) are the six trigonometric ratios (sec).
