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CpE Maj
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This may be basic, but how do you know which part of an integration problem fill the up the triangle values for trig substitution?
Trigonometric Substitution Triangles
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In summary, a trigonometric substitution triangle is a right triangle used in calculus to simplify integrals with trigonometric functions. It is created by identifying the function in the integral, choosing a trigonometric substitution, and using the Pythagorean theorem. These triangles can make solving integrals easier and more efficient, but they have limitations and may not always provide the most straightforward solution. It is important to understand when and how to use them properly.
This may be basic, but how do you know which part of an integration problem fill the up the triangle values for trig substitution?
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tiny-tim
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Welcome to PF!
Hi CpE Maj! Welcome to PF!
I don't understand.
Hi CpE Maj! Welcome to PF!
CpE Maj said:
I don't understand.
Can you give an example?
1. What is a trigonometric substitution triangle?
A trigonometric substitution triangle is a right triangle that is used in calculus to simplify integrals involving trigonometric functions. It is made up of three sides, one of which is the hypotenuse, and three angles, one of which is a right angle.
2. When should I use trigonometric substitution triangles?
Trigonometric substitution triangles are used to simplify integrals that involve certain types of functions, such as radicals, polynomials, or algebraic expressions. They are especially useful when dealing with integrals that contain both algebraic and trigonometric terms.
3. How do I create a trigonometric substitution triangle?
To create a trigonometric substitution triangle, you first need to identify the type of function in the integral. Then, choose an appropriate trigonometric substitution, such as sine, cosine, or tangent, and assign one of the triangle's angles to that substitution. Finally, use the Pythagorean theorem to determine the remaining sides of the triangle.
4. What are the benefits of using trigonometric substitution triangles?
Using trigonometric substitution triangles can make solving integrals much easier and more efficient. They allow you to replace complicated algebraic expressions with simpler trigonometric functions, making the integral more manageable. Additionally, they can help you to recognize patterns in integrals and apply appropriate trigonometric identities.
5. Are there any limitations to using trigonometric substitution triangles?
Trigonometric substitution triangles can only be used for certain types of integrals, such as those that contain algebraic and trigonometric terms. They are not applicable to all integrals, so it's important to understand when and how to use them properly. Additionally, they may not always provide the most straightforward solution, so it's important to consider other methods as well.
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