Understanding Trig Quadrants and the Role of |k| ≥ 1 in Solving for θ

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In summary, the conversation discusses how to determine the quadrant of an angle θ based on given information. It is mentioned that θ is obtuse, which eliminates the first quadrant as a possibility. The solutions state that θ is in the second quadrant, with k < 0. However, there is a question about whether θ could also be in the third or fourth quadrant. The purpose of the equation |k|≥ 1 is also questioned. It is then clarified that obtuse angles are between 90 and 180 degrees, and that angles between 180 and 360 degrees are called reflex. The purpose of |k|≥ 1 is confirmed to enforce that θ is a real angle.
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I'm just curious on how to find out what quadrant θ is in from the information. I know it says that theta is obtuse, but doesn't this only conclude that theta is not in the first quadrant? In the solutions they have right away said theta is obtuse therefore it is in the second quadrant so k is < 0, which I can understand but couldn't theta be in the 3rd & 4th quadrant also? Also what is |k|≥ 1 used for?

EDIT: nevermind, I though obtuse was just >90, not between 90 and 180 :x - If anyone could answer what |k| is for it'd help thanks.
 
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  • #2
The name for angles that are between 180 and 360 are called reflex.

And as for the [itex]|k|\geq 1[/itex] this is probably just enforcing that [itex]\theta[/itex] is real.
 
  • #3
Mentallic said:
The name for angles that are between 180 and 360 are called reflex.

And as for the [itex]|k|\geq 1[/itex] this is probably just enforcing that [itex]\theta[/itex] is real.

thanks again.
 

Related to Understanding Trig Quadrants and the Role of |k| ≥ 1 in Solving for θ

1. What are the four quadrants in trigonometry?

The four quadrants in trigonometry are the first quadrant (I), second quadrant (II), third quadrant (III), and fourth quadrant (IV). These quadrants are formed by the x-axis and y-axis, and they help determine the sign of the trigonometric functions based on the coordinates of the angle.

2. How do you determine the quadrant of an angle?

To determine the quadrant of an angle, you need to look at the signs of the x and y coordinates of the angle. If both coordinates are positive, the angle lies in the first quadrant (I). If the x-coordinate is negative and the y-coordinate is positive, the angle is in the second quadrant (II). If both coordinates are negative, the angle is in the third quadrant (III). And if the x-coordinate is positive and the y-coordinate is negative, the angle lies in the fourth quadrant (IV).

3. What is the role of |k| ≥ 1 in solving for θ?

The role of |k| ≥ 1 in solving for θ is to determine the number of solutions for the trigonometric equation. If |k| ≥ 1, there will be two solutions for θ, while if |k| < 1, there will be no solution. This is because |k| represents the amplitude of the trigonometric function, and if it is larger than or equal to 1, the function will intersect the x-axis twice within one period, resulting in two solutions for θ.

4. How do you solve for θ in a trigonometric equation with |k| ≥ 1?

To solve for θ in a trigonometric equation with |k| ≥ 1, you need to use the inverse trigonometric functions (arcsine, arccosine, and arctangent). First, isolate the trigonometric function on one side of the equation. Then, take the inverse of the trigonometric function on both sides to eliminate the function. This will give you an equation in terms of θ. Finally, use a calculator or reference table to find the numerical value of θ.

5. Can an angle be in multiple quadrants?

No, an angle can only be in one quadrant. This is because the quadrant is determined by the sign of the coordinates, and an angle cannot have both positive and negative coordinates at the same time. However, an angle can have multiple reference angles in different quadrants, which are the acute angles formed between the terminal side of the angle and the x-axis.

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