Question
Why is there always a positive and negative solution when using the Pythagorean identity
Answers
GPT-4o mini
The Pythagorean identity states that for any angle \(\theta\):
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
This identity implies that the values of \(\sin(\theta)\) and \(\cos(\theta)\) can be represented as coordinates on the unit circle, where the radius (the hypotenuse) is always 1.
When you solve for \(\sin(\theta)\) or \(\cos(\theta)\) from this identity, such as rearranging it to find \(\sin(\theta)\):
\[
\sin(\theta) = \sqrt{1 - \cos^2(\theta)} \quad \text{or} \quad \sin(\theta) = -\sqrt{1 - \cos^2(\theta)}
\]
you get two solutions because both the positive and negative square roots are valid. This reflects the fact that the sine function can take both positive and negative values depending on the quadrant in which the angle \(\theta\) lies:
- In the first quadrant (0 to 90 degrees), both \(\sin(\theta)\) and \(\cos(\theta)\) are positive.
- In the second quadrant (90 to 180 degrees), \(\sin(\theta)\) is positive and \(\cos(\theta)\) is negative.
- In the third quadrant (180 to 270 degrees), both \(\sin(\theta)\) and \(\cos(\theta)\) are negative.
- In the fourth quadrant (270 to 360 degrees), \(\sin(\theta)\) is negative and \(\cos(\theta)\) is positive.
So whenever you take the square root of a trigonometric function when using the Pythagorean identity, you should consider both the positive and negative values as valid solutions, depending on the quadrant where the angle is located. This is why there are always both a positive and a negative solution when working with the Pythagorean identity in the context of trigonometric functions.
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
This identity implies that the values of \(\sin(\theta)\) and \(\cos(\theta)\) can be represented as coordinates on the unit circle, where the radius (the hypotenuse) is always 1.
When you solve for \(\sin(\theta)\) or \(\cos(\theta)\) from this identity, such as rearranging it to find \(\sin(\theta)\):
\[
\sin(\theta) = \sqrt{1 - \cos^2(\theta)} \quad \text{or} \quad \sin(\theta) = -\sqrt{1 - \cos^2(\theta)}
\]
you get two solutions because both the positive and negative square roots are valid. This reflects the fact that the sine function can take both positive and negative values depending on the quadrant in which the angle \(\theta\) lies:
- In the first quadrant (0 to 90 degrees), both \(\sin(\theta)\) and \(\cos(\theta)\) are positive.
- In the second quadrant (90 to 180 degrees), \(\sin(\theta)\) is positive and \(\cos(\theta)\) is negative.
- In the third quadrant (180 to 270 degrees), both \(\sin(\theta)\) and \(\cos(\theta)\) are negative.
- In the fourth quadrant (270 to 360 degrees), \(\sin(\theta)\) is negative and \(\cos(\theta)\) is positive.
So whenever you take the square root of a trigonometric function when using the Pythagorean identity, you should consider both the positive and negative values as valid solutions, depending on the quadrant where the angle is located. This is why there are always both a positive and a negative solution when working with the Pythagorean identity in the context of trigonometric functions.