Trigonometric identities are equations that involve trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant, and are true for all values of the variable(s) involved. They can be used to simplify trigonometric expressions, solve equations, and prove other identities.
Trigonometric identities are derived from the properties of the trigonometric functions and the relationships between them. These identities can be divided into several categories, including:
- Pythagorean identities: These identities involve the relationship between the sine and cosine of an angle in a right triangle.
- Reciprocal identities: These identities involve the reciprocal functions (cosecant, secant, and cotangent) and relate them to the sine and cosine functions.
- Quotient identities: These identities involve the relationship between the trigonometric functions in the form of ratios.
- Product-to-sum and sum-to-product identities: These identities involve expressing the product or sum of two trigonometric functions in terms of other trigonometric functions.
- Double angle and half angle identities: These identities involve expressing trigonometric functions of double or half an angle in terms of trigonometric functions of the original angle.
These identities can help you simplify trigonometric expressions in equations and allow you to express complex trigonometric equations in simpler forms. And also helpful in solving trigonometry problems in various fields like Physics, engineering, navigation etc.
Sum and Difference of Angles Trigonometric Identities
The sum and difference of angles trigonometric identities are identities that express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. These identities are useful when you need to add or subtract two angles in an equation.
Here are the most common sum and difference identities:
- Sum-to-product identities: These identities express the sine and cosine of the sum of two angles in terms of the sines and cosines of the individual angles.
- sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
- Difference-to-product identities: These identities express the sine and cosine of the difference of two angles in terms of the sines and cosines of the individual angles.
- sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
- cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
where “a” and “b” are the two angles.
The sum-to-product and difference-to-product identities can be helpful in solving trigonometry problems, such as finding the value of a trigonometric expression that involves the sum or difference of two angles. For example, they can be used to solve trigonometry equations involving two angles.
These identities are also particularly useful in the process of integration. It can help to transform a product of trigonometric functions into the sum or difference of trigonometric functions which are much easier to integrate.
Trigonometric Identities Formulas
Double Angle Trigonometric Identities
The double angle trigonometric identities are identities that express trigonometric functions of double the angle in terms of trigonometric functions of the original angle. These identities are useful when you need to express an angle in terms of double of it or when you want to simplify an expression involving an angle that is double another.
Here are the most common double angle identities:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos^2(θ) – sin^2(θ) = 1 – 2sin^2(θ) = 2cos^2(θ) – 1 = 1 – 2sin^2(θ)
- tan(2θ) = (2tan(θ)) / (1 – tan^2(θ))
where “θ” is the angle.
These identities can be derived by using the sum-to-product identities and the Pythagorean identity. The double angle formula can be derived by sum-to-product identity, you can write sin(2θ) = sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ) Similarly, The double angle formula for cos(2θ) can be derived using cos(2θ) = cos(θ + θ) = cos(θ)cos(θ) – sin(θ)sin(θ) = cos^2(θ) – sin^2(θ)
These identities can help you simplify trigonometric expressions that involve double angles, and also can be used to solve trigonometry equations that involve double angles. They are also useful in calculus, for example in integration and differentiation when some angles in the function are in double angle form.
Half Angle Identities
The half angle trigonometric identities are identities that express trigonometric functions of half an angle in terms of trigonometric functions of the original angle. These identities are useful when you need to express an angle in terms of half of it or when you want to simplify an expression involving an angle that is half another.
Here are the most common half angle identities:
- sin(θ/2) = ±√[(1 – cos(θ)) / 2]
- cos(θ/2) = ±√[(1 + cos(θ)) / 2]
- tan(θ/2) = ±√[(1 – cos(θ)) / (1 + cos(θ))]
where “θ” is the angle, the sign for the square root is positive if 0 < θ/2 < π and negative if π < θ/2 < 2π.
These identities can be derived using the sum-to-product identities and the double angle identities. For example, you can use the double angle identity for sine, sin(2θ) = 2sin(θ)cos(θ) to find the half angle identity, sin(θ/2) = ± √[sin(2θ)/2] = ±√[(1 – cos(θ)) / 2].
These identities are useful in simplifying trigonometric expressions that involve half angles, and also can be used to solve trigonometry equations that involve half angles. They are also helpful in calculus, for example, in integration and differentiation when some angles in the function are in half angle form. It is also useful in solving certain types of geometric problems and certain types of wave propagation problems and it is also used in certain types of elliptic integrals.
Product-Sum Trigonometric Identities
Product-sum identities are trigonometric identities that express the product of two trigonometric functions in terms of the sum or difference of the two functions. These identities are useful for simplifying trigonometric expressions and for solving trigonometry problems that involve products of trigonometric functions.
Here are the most common product-sum identities:
- Product-to-sum identities: These identities express the product of two trigonometric functions in terms of the sum of the two functions. For example:
- sin(a)cos(b) = (1/2)[sin(a+b) + sin(a-b)]
- cos(a)cos(b) = (1/2)[cos(a-b) + cos(a+b)]
- sin(a)sin(b) = (1/2)[cos(a-b) – cos(a+b)]
- Sum-to-product identities: These identities express the sum of two trigonometric functions in terms of the product of the two functions. For example:
- sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
where “a” and “b” are the two angles.
The product-to-sum identities are useful in solving trigonometry problems that involve products of trigonometric functions, for example in finding the value of an expression that contains a product of sine and cosine of different angles. The sum-to-product identities are useful when you have the sum of trigonometric functions, and you want to express it in terms of product, it can help you to simplify trigonometric expressions.
These identities are also useful in calculus, for example in integration, these identities help to transform a product of trigonometric functions into the sum or difference of trigonometric functions which are much easier to integrate. In some cases of solving physical problems, like wave propagation and quantum mechanics, it’s important to use the product-sum identities to simplify the problem.
Trigonometric Identities of Products
Trigonometric identities of products are identities that express the product of two or more trigonometric functions in terms of other trigonometric functions. These identities can be used to simplify trigonometric expressions and to solve equations that involve products of trigonometric functions.
Here are some examples of trigonometric identities of products:
- Product of sine and cosine:
- sin(x)cos(y) = (1/2)[sin(x+y) + sin(x-y)]
- cos(x)sin(y) = (1/2)[sin(x+y) – sin(x-y)]
- cos^2(x) = (1/2)(1 + cos(2x))
- sin^2(x) = (1/2)(1 – cos(2x))
- Product of tangent and cotangent:
- tan(x)cot(y) = (1/2)[cot(x+y) – cot(x-y)]
- cot(x)tan(y) = (1/2)[cot(x+y) + cot(x-y)]
- cot^2(x) = csc^2(x) – 1
- tan^2(x) = sec^2(x) – 1
- Product of sine and tangent:
- sin(x)tan(y) = (1/2)[sin(x+y) – sin(x-y)]
- sin(x)cot(y) = (1/2)[cos(x-y) – cos(x+y)]
- Product of cosine and secant:
- cos(x)sec(y) = (1/2)[sec(x+y) + sec(x-y)]
- cos(x)csc(y) = (1/2)[csc(x+y) – csc(x-y)]
- Product of cosecant and cotangent
- csc(x)cot(y) = (1/2)[csc(x+y) + csc(x-y)]
These above identities can be used to simplify trigonometric expressions that involve products of trigonometric functions and also to solve trigonometry equations that involve products of trigonometric functions. These identities are very useful in many fields like physics, engineering, navigation and so on. It is important to note that for some of these identities, it is required that x+y and x-y should be between -π/2 to π/2 for some of the above identities to work.
Trigonometric Identities Class 10
In class 10, students typically learn the following trigonometric identities:
- Pythagorean identities: These identities express the relationship between the sine and cosine of an angle in a right triangle. The two identities are: sin^2(x) + cos^2(x) = 1 and 1 + tan^2(x) = sec^2(x).
- reciprocal identities: These identities express the reciprocal of trigonometric functions in terms of other trigonometric functions. For example: cosec(x) = 1/sin(x), sec(x) = 1/cos(x), and cot(x) = 1/tan(x).
- quotient identities: These identities express the quotient of trigonometric functions in terms of other trigonometric functions. For example: tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x).
- co-function identities: These identities express the sine and cosine of an angle in terms of the tangent and cotangent of the complement angle. For example: sin(90 – x) = cos(x) and cos(90 – x) = sin(x).
- double angle identities: These identities express trigonometric functions of double the angle in terms of trigonometric functions of the original angle. For example: sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) – sin^2(x).
- half angle identities: These identities express trigonometric functions of half an angle in terms of trigonometric functions of the original angle. For example: sin(x/2) = ± √[(1 – cos(x)) / 2] and cos(x/2)
Trigonometric Identities Class 11
In class 11, students typically learn more advanced trigonometric identities, building on what they learned in class 10. Some examples of identities that may be covered in class 11 include:
- Product-to-Sum identities: These identities express the product of two trigonometric functions in terms of the sum or difference of the two functions. For example: sin(a)cos(b) = (1/2)[sin(a+b) + sin(a-b)] and cos(a)sin(b) = (1/2)[sin(a+b) – sin(a-b)].
- Sum-to-Product identities: These identities express the sum of two trigonometric functions in terms of the product of the two functions. For example: sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
- Identity involving the square of trigonometric functions: These identities express the square of trigonometric functions in terms of the product of the same function and another function. For example: sin^2(x) = (1-cos(2x))/2, cos^2(x) = (1+cos(2x))/2, and tan^2(x) = (sec^2(x) -1).
- Product-Quotient Identities : These identities express the product or quotient of two trigonometric functions in terms of other trigonometric functions. For example : sin(x)cos(y) = (1/2)[sin(x+y) – sin(x-y)] and tan(x) + cot(x) = sec^2(x)
Solved Examples on Trigonometric Identities
These examples show how to use trigonometric identities to simplify expressions and equations that involve trigonometric functions. These identities are powerful tools for solving trigonometry problems, and it is essential to memorize them and be able to use them correctly.
here are a few examples of solving trigonometric problems using trigonometric identities:
Example 1: Simplify the expression: sin(x)cos(x)
Solution: We can use the identity sin(x)cos(x) = (1/2)[sin(2x)], where sin(2x) = 2sin(x)cos(x) So, the expression becomes (1/2)[2sin(x)cos(x)] = (1/2)sin(x)cos(x)
Example 2: Simplify the expression: cos^2(x) – sin^2(x)
Solution: We can use the identity cos^2(x) – sin^2(x) = cos(2x) , where cos(2x) = cos^2(x) – sin^2(x) so the expression becomes cos(2x) = cos^2(x) – sin^2(x)
Example 3: Simplify the expression: cot(x)cot(y)
Solution: We can use the identity cot(x)cot(y) = csc^2(x) – csc^2(y), where csc^2(x) = 1/sin^2(x) and csc^2(y) = 1/sin^2(y) so the expression becomes cot(x)cot(y) = (1/sin^2(x)) – (1/sin^2(y))
Example 4: Simplify the expression: tan(x) – sec^2(x)
Solution: We can use the identity tan(x) = sec^2(x) – 1, where sec^2(x) = 1/cos^2(x) so the expression becomes tan(x) – sec^2(x) = 1 – sec^2(x) – 1 = -sec^2(x) + 1
Example 5: Simplify the expression: sin(x+y) + sin(x-y)
Solution: We can use the identity sin(x+y) + sin(x-y) = 2sin(x)cos(y) so the expression becomes sin(x+y) + sin(x-y) = 2sin(x)cos(y)
Frequently Asked Questions on Trigonometric Identities
These are just a few examples of the many ways that trigonometric identities are used in real-world problems. It’s important to practice using trigonometric identities to become comfortable with them and to be able to use them effectively in solving problems.
What is the difference between a trigonometric identity and an equation?
A trigonometric identity is an equation that is true for all values of the variable(s) involved, while an equation is a statement that asserts that two expressions are equal, but only for certain values of the variable(s) involved.
How do I use trigonometric identities to solve equations?
To solve an equation using a trigonometric identity, you first use the identity to simplify one or both sides of the equation, and then use algebraic methods to solve for the variable(s) involved.
What are the Pythagorean identities?
The Pythagorean identities are a set of trigonometric identities that express the relationship between the sine and cosine of an angle in a right triangle. The two identities are: sin^2(x) + cos^2(x) = 1 and 1 + tan^2(x) = sec^2(x).
How are double angle identities derived?
Double angle identities are derived by using the sum-to-product identities and the Pythagorean identity. For example, the double angle formula for sine can be derived by using the sum-to-product identity sin(2x) = sin(x + x) = sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x).
What is the difference between sum-to-product and product-to-sum identities?
Sum-to-product identities express the sum of two trigonometric functions in terms of the product of the two functions. Product-to-sum identities express the product of two trigonometric functions in terms of the sum of the two functions.
How do I know when to use a particular identity?
To use a particular identity, you need to first recognize the trigonometric functions and the angles involved in the problem and then match them with the appropriate identities. Practice and familiarity with the identities will make it easier to recognize when to use a particular identity.
