# Trig and double angle integration

• banshee43
In summary, trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The unit circle is a circle with a radius of 1 unit that is used to represent the values of trigonometric functions. The double angle formula is a trigonometric identity that allows us to express the sine, cosine, or tangent of a double angle in terms of the sine, cosine, or tangent of the original angle. This formula can be used to simplify integrals involving trigonometric functions, making it useful in various real-life applications such as physics, engineering, and navigation.
banshee43

∫sin(t)cos(2t)dt

## Homework Equations

cos(2t)=1-2sin^2(t)

## The Attempt at a Solution

∫sint(t)(1-2sin^2(t))dt
but where do i go from here? i can't use substitution. or parts?

cos2x can also = 2cos2x-1, if you make it that, then this is a simple u-substitution..
u=cosx

thankyou very much! i just didnt exhaust my formulas..

## 1. What is a trigonometric function?

A trigonometric function is a mathematical function that relates the angles of a triangle to the lengths of its sides. The most commonly used trigonometric functions are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

## 2. What is the unit circle and how is it related to trigonometric functions?

The unit circle is a circle with a radius of 1 unit, centered at the origin of a graph. It is used to represent the values of trigonometric functions for any given angle. The x-coordinate of a point on the unit circle represents the cosine of the angle, while the y-coordinate represents the sine of the angle.

## 3. What is the double angle formula?

The double angle formula is a trigonometric identity that allows us to express the sine, cosine, or tangent of a double angle in terms of the sine, cosine, or tangent of the original angle. The formulas are as follows:
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)
tan(2θ) = 2tan(θ) / (1 - tan²(θ))

## 4. How can the double angle formula be used to simplify integrals?

The double angle formula can be used in integration to simplify integrals involving trigonometric functions. By using the double angle formula to rewrite the trigonometric functions in terms of the original angle, the integral may become easier to solve. This is especially useful when dealing with integrals involving powers of sine or cosine functions.

## 5. What are some real-life applications of trigonometric and double angle integration?

Trigonometric and double angle integration have many real-life applications, including in physics, engineering, and navigation. For example, trigonometric functions are used to model the motion of waves and vibrations, while the double angle formula is used in calculating the stress and strain in structures. In navigation, the unit circle and trigonometric functions are used to calculate distances and angles between objects, as well as in GPS systems and satellite communication.

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