Each point lies on the terminal arm of an angle θin standard position. Determine θin the specifi

Each point lies on the terminal arm of an angle θin standard...

Key Concepts

Standard Position

An angle is said to be in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. This concept is fundamental in coordinate geometry because it provides a common reference for measuring angles, making it possible to associate each point on the terminal arm of an angle with a unique measure determined by its position relative to the positive x-axis.

Terminal Arm and Quadrants

The terminal arm of an angle in standard position is the ray that results after the angle's rotation from the positive x-axis. Where this arm lies determines the quadrant in which the end point is located. Understanding the quadrant is essential because it affects the sign of the coordinates and informs the necessary adjustments when using inverse trigonometric functions to determine the angle's measure.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to calculate the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of a point. However, since the arctan function typically returns values in a restricted interval, it is important to adjust the calculated angle to reflect the actual quadrant in which the terminal arm is situated.

Angle Measurement and Domain Restrictions

After determining an angle using inverse trigonometric functions, it is often necessary to express the angle in a specified domain, which might include adding or subtracting full rotations (360° or 2? radians) to obtain an equivalent angle that falls within the required interval. This ensures that the angle measure is meaningful in the given context and adheres to any stipulated boundaries.

Rounding to the Nearest Hundredth

When precise values are required, especially in applications or standardized assessments, the computed angle measure may need to be rounded to the nearest hundredth. This involves approximating the result to two decimal places, which provides a balance between precision and simplicity, particularly when results are obtained from inverse trigonometric functions that yield irrational numbers.

Transcript

00:03 So number 19 asks you to find theta if theta is the angle of the line segment that passes through the point negative 3 -4.

00:15 So negative 3 -4 is in quadrant 2, and i need to find a theta for that.

00:24 And theta needs to be in radiance between 0 and 4 -pi.

00:28 So we're going to use theta is 10 inverse of y over x.

00:35 You're going to set the calculator in radiance, and the calculator is going to always give you an angle between negative pi over 2 and pi over 2.

00:45 So it's going to give me an answer in quadrant 4 to start with.

00:49 When you plug that in, you get negative 0 .927, which isn't one of the answers we want, but it does get you started.

00:58 The ones in quadrant 2 are pi radians away from this one, so the theta's i'm allowed to use, we'll start at negative .97 plus pi.

01:13 So the first one we get is 2 .21 radiance.

01:18 That is in quadrant 2.

01:21 And since they gave me 0 to 4 pi, they want the first one here, and the second one all the way around and back to here.

01:29 So take 2 .21 and add 2 pi to it.

01:34 You get the second angle that works, which is 8 .50.

01:39 Letter b, wants you to look at the 0 .5 negative 1, which is in quadrant 4.

01:49 We're in degrees.

01:50 You have to set the calculator in degrees before you start finding the angle.

01:56 Theta is tan inverse of the y over the x, and you get theta is negative 11.

02:09 They want all the values from negative 360 around a positive 360.

02:16 Over 5 and down 1 gives me negative 310.

02:21 If i subtract 360, the next one would be too small.

02:26 So i'm going to write down my list of correct ones, negative 11 .310 degrees.

02:33 And then add 360 to that, and you get 348 .69...

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