91影视

The Pythagorean Identity is a fundamental concept in trigonometry that connects the squares of the sine and cosine of an angle. It is an equation that states:

• \( \sin^2\theta + \cos^2\theta = 1 \)

This identity is derived from the Pythagorean theorem when considering the unit circle (a circle with radius 1). The idea is that any point on the unit circle represents the sine and cosine of an angle where the x-coordinate is \( \cos\theta \) and the y-coordinate is \( \sin\theta \).
By rearranging and solving this equation, you can find one trigonometric function if the other is known, along with the quadrant information to determine the sign.
In our problem, we knew \( \cos\theta = -\frac{7}{12} \). Using the Pythagorean Identity:

• \( \sin^2\theta = 1 - \left(-\frac{7}{12}\right)^2 = \frac{95}{144} \)

To find \( \sin\theta \), we take the square root on both sides, considering that in Quadrant III, sine is negative:

• \( \sin\theta = -\frac{\sqrt{95}}{12} \)

Sine and cosine are the primary trigonometric functions that describe the relationship between the sides of a right triangle and its angles. They are also fundamental in describing the movement on a unit circle.
The relationship between sine and cosine is most elegantly captured by the Pythagorean Identity discussed earlier. However, to find other functions like tangent, we utilize another relationship:

• \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)

This equation shows that tangent is the ratio of sine to cosine, making it easy to calculate when both are known.
In our example, after finding \( \sin\theta = -\frac{\sqrt{95}}{12} \) and given \( \cos\theta = -\frac{7}{12} \), we find:

• \( \tan\theta = \frac{-\frac{\sqrt{95}}{12}}{-\frac{7}{12}} = \frac{\sqrt{95}}{7} \)

Here, the negatives cancel each other out, and we see tangent is positive in Quadrant III.

Understanding the quadrant in which an angle resides is crucial for determining the signs of trigonometric functions. In trigonometry, a circle is divided into four quadrants based on the x and y axes.
Quadrant III is the bottom left section of this circle:

• Here, cosine and sine values are both negative because the x and y coordinates in this quadrant are negative.
• However, tangent, being the division of sine by cosine, becomes positive since a negative divided by a negative is a positive.

Knowing this helps us determine that in our problem, when \( \cos\theta = -\frac{7}{12} \), \( \sin\theta \) should be negative. We calculated \( \sin\theta = -\frac{\sqrt{95}}{12} \).
For tangent, \( \tan\theta = \frac{\sqrt{95}}{7} \), verifying positivity, which fits the characteristic of Quadrant III. 协褌芯褌 邪薪邪谢懈蟹 褌邪泻卸械 芯斜谢械谐褔邪械褌 褉邪褋褔械褌 芯斜褉邪褌薪褘褏 褎褍薪泻褑懈泄, 褌邪泻懈褏 泻邪泻 泻邪褋邪褌械谢褜薪邪褟, 褋械泻邪薪褋 懈 泻芯-褋械泻邪薪褋, 薪邪 芯褋薪芯胁械 懈褏 芯褋薪芯胁薪褘褏 褎褍薪泻褑懈泄.