Find sin2 x, cos2 x, and tan2 x from the given information. sinx=-(3)/(5), x in quadrant III

Name: Problem 5, Chapter 7: Precalculus: Mathematical for Calculus
Uploaded: 2023-10-15T02:34:30-08:00
Duration: 4 min 22 s
Channel: Marcella Sippey
Description: Find sin2 x, cos2 x, and tan2 x from the given information. sinx=-(3)/(5), x in quadrant III

Find sin2 x, cos2 x, and tan2 x from the given information....

Key Concepts

Double Angle Identities

Double angle identities express trigonometric functions of twice an angle in terms of the functions of the original angle. These identities, such as sin(2x) = 2 sin(x) cos(x) and cos(2x) = cos²(x) - sin²(x), are valuable tools in solving problems where the angle is doubled or manipulated, streamlining the process of simplifying or evaluating expressions.

Pythagorean Identity

The Pythagorean identity, sin²(x) + cos²(x) = 1, is a fundamental relationship in trigonometry that connects the sine and cosine functions. It is particularly useful for finding an unknown trigonometric value when one of the functions is given, ensuring that the values are consistent with the geometric properties of a right triangle or unit circle.

Quadrant Considerations

Quadrant considerations are essential when working with trigonometric functions because the sign (positive or negative) of these functions depends on the quadrant in which the angle lies. Correctly applying these sign conventions is crucial when solving problems, as it helps determine the appropriate values of sine, cosine, and tangent based on the angle's location in the coordinate system.

Transcript

00:01 All right, this problem wants us to find the sine of 2x, pen work, there it goes, find the sine of 2x, cosine of 2x and tangent of 2x from the given information.

00:24 And they're telling us the sine of x is equal to negative 3 5ths and that x is in quadrant three.

00:36 So what we need first is a really nice unit circle.

00:44 And i wanna make sure that i put my angle down here in quadrant three.

00:52 Now from right triangle trig, we should know soh cah toa.

00:58 And that of course stands for sine is opposite over hypotenuse.

01:03 So since theta is here, opposite is going to be negative 3, hypotenuse is gonna be 5.

01:10 And then we should recognize this as a 3 4 5 triangle.

01:14 So that's 4.

01:16 And from there we can find the cosine is adjacent over hypotenuse, so 4 over 5.

01:23 And our tangent is going to be opposite over adjacent, so negative 3 over 4.

01:31 And we need all of those things because now we have to find our double angles.

01:36 And i brought over the double angle identities already.

01:39 So when we are finding the sine of 2x, we just need to take two times the sine of x times the cosine of x.

01:50 Well, we know the sine of x is of course negative 3 5ths.

01:55 And we know the cosine of x is 4 5ths.

01:59 So this just becomes two times negative 3, which is negative 6 times 4, which is a negative 24 over 5 times 5, which is 25.

02:09 So the sine of 2x is negative 24 25ths...

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