Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x

Question

Very confused! Please help!

sin ( tan^-1 2x )

Answer 1

Let θ = tan^-1(2x)

Therefore:

tan θ = 2x or tan θ = 2x/1

Remember the definition of tangent? tan θ = _(Opposite)_ / (Adjacent).

So if the two legs of a right triangle are 2x and 1, then we can obtain the hypotenuse using Pythagorean

_________ _________

Theorem which is √ (2x)² + 1^{2 =} √ 4x² + 1

⁄|

⁄ |

⁄ |

⁄ |

______ ⁄ |

√ 4x²+1 ⁄ |

⁄ | 2x

⁄ |

⁄ |

⁄ θ |

⁄________ ______⌋

1

sin (tan^-1(2x)) = sin θ = ___2x___

√4x² + 1

because sine is defined as opposite over hypotenuse.