Trigonometry is the study of relationships that deal with angles, lengths and heights of triangles and relations between different parts of circles and other geometrical figures. Applications of trigonometry are also found in engineering, astronomy, Physics and architectural design.
Trigonometric identities are very useful and learning the below formulae help in solving the problems better. There is an enormous number of fields where these identities of trigonometry and formula of trigonometry are used.
Now to get started let us start with noting the difference between Trigonometric identities and Trigonometric Ratios.
- Trigonometric Identities are some formulas that involve the trigonometric functions. These trigonometry identities are true for all values of the variables.
- Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the side of the right triangle.
Learn more about Trigonometric Ratios here in detail.
Now let us start with the basic formulas of trigonometry and see the basic relationships on which the whole concept is based on.
In a right-angled triangle, we have Hypotenuse, Base and Perpendicular. The longest side is known as the hypotenuse, the other side which is opposite to the angle is Perpendicular and the third side is Base. The six trigonometric functions are sine, cosine, secant, cosecant, tangent and cotangent. So now all the trigonometric ratios are based on the lengths of these lengths of the side of the triangle and the angle of the triangle.
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The Trigonometric properties are given below:
The reciprocal relationships between different ratios can be listed as:
The basic trigonometric identities based on the Pythagoras Theorem are listed here. You can use the basic definition and Pythagoras theorem to prove these.
Trigonometric ratios for negative angles can be derived using the circular concept of negative angles and can be derived using cartesian notation and conventions.
Periodicity and Periodic Identities
The basic concept of trigonometry is based on the repetition of the values of sine, cos and tan after 360⁰ due to their periodic nature.
If n is an integer and in radians (if in degrees the replace with 360)
If the angles are given in any of the four quadrants then the angle can be reduced to the equivalent first quadrant by changing signs and trigonometric ratios:
Sum to product rules
Some identities related to sum and difference of two angles can be listed as follows:
Product to sum rules
And from the above identities we can derive the multiplication rules and the summation rules of two trigonometric ratios:
Double angle identities
We can get the double angle identities if we put y = x in the above equations and can get the following identities:
Half angle identities
Now using the above equations, we can get the half angle relations by putting x = x/2 and using all the identities we can derive the following:
The trigonometric equations can also be related to complex numbers and through the following relations:
Inverse trigonometric functions
Now let us talk about inverse trigonometric relations. They are sometimes denoted with a -1 in the superscript of the trigonometric ratios and sometimes also denoted using arc as a prefix, for example, sin-1, cos-1, arctan etc.
They have their own properties and though they are periodic like sine, cos, tan they have a convention while solving problems:
And they have relations with trigonometric functions as listed below:
Their other properties include: