Finding Cosine Given Sine And Quadrant Argue Theta Is In Standard Position

Hey guys! Let's dive into a fun trigonometry problem where we need to figure out the cosine of an angle, given its sine and the quadrant it lives in. This is a classic problem that combines our knowledge of trigonometric identities and the unit circle, so let's break it down step by step.

Understanding Standard Position and the Unit Circle

Before we jump into the problem, let's make sure we're all on the same page about standard position and the unit circle. An angle, often denoted by the Greek letter theta ($\theta$), is said to be in standard position when its initial side lies along the positive x-axis and its vertex is at the origin (0,0) of the Cartesian plane. The terminal side is the ray that rotates away from the initial side, and the angle of rotation is our angle $\theta$.

Now, picture a circle with a radius of 1 centered at the origin – this is our trusty unit circle. Any point on this circle can be represented by coordinates (x, y), and these coordinates are directly related to the cosine and sine of the angle formed by the positive x-axis and the line connecting the origin to that point. Specifically, the x-coordinate is the cosine of the angle, and the y-coordinate is the sine of the angle. This is a fundamental concept in trigonometry, so make sure you're comfortable with it.

Key takeaway: On the unit circle, x = cos(θ) and y = sin(θ). This simple relationship is super powerful and will help us solve many trig problems.

Quadrants and Trigonometric Signs

The unit circle is divided into four quadrants, and the signs of sine and cosine vary in each quadrant. This is where the mnemonic "All Students Take Calculus" (ASTC) comes in handy. It helps us remember which trigonometric functions are positive in each quadrant:

• Quadrant I (0 < θ < π/2): All trigonometric functions (sine, cosine, tangent, etc.) are positive.
• Quadrant II (π/2 < θ < π): Sine is positive (and its reciprocal, cosecant).
• Quadrant III (π < θ < 3π/2): Tangent is positive (and its reciprocal, cotangent).
• Quadrant IV (3π/2 < θ < 2π): Cosine is positive (and its reciprocal, secant).

This information is crucial because it tells us the sign of our cosine value in this specific problem. Knowing the quadrant in which our angle lies helps us determine whether the cosine should be positive or negative.

In our case, we're told that $\pi < \theta < \frac{3\pi}{2}$, which means our angle $\theta$ is in Quadrant III. According to ASTC, only tangent is positive in Quadrant III, so both sine and cosine are negative. This is a vital piece of the puzzle!

The Problem: Finding cos(θ)

Alright, let's get to the heart of the problem. We're given that $\sin(\theta) = -\frac{1}{3}$ and $\pi < \theta < \frac{3\pi}{2}$. Our mission, should we choose to accept it, is to find $\cos(\theta)$.

Using the Pythagorean Identity

The key to solving this lies in the Pythagorean identity, one of the most fundamental trigonometric identities:

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

This identity is derived directly from the Pythagorean theorem applied to the unit circle. It essentially states that the sum of the squares of the sine and cosine of any angle is always equal to 1. This is a powerhouse identity that we'll use extensively.

We already know $\sin(\theta)$, so we can plug that value into the Pythagorean identity and solve for $\cos(\theta)$:

$\left(-\frac{1}{3}\right)^2 + \cos^2(\theta) = 1$

Simplify the equation:

$\frac{1}{9} + \cos^2(\theta) = 1$

Now, subtract $\frac{1}{9}$ from both sides:

$\cos^2(\theta) = 1 - \frac{1}{9}$

$\cos^2(\theta) = \frac{8}{9}$

Solving for cos(θ) and Considering the Quadrant

To find $\cos(\theta)$, we need to take the square root of both sides:

$\cos(\theta) = \pm\sqrt{\frac{8}{9}}$

$\cos(\theta) = \pm\frac{\sqrt{8}}{3}$

$\cos(\theta) = \pm\frac{2\sqrt{2}}{3}$

Now, here's where our knowledge of quadrants comes into play. We determined earlier that $\theta$ is in Quadrant III, where cosine is negative. Therefore, we must choose the negative solution:

$\cos(\theta) = -\frac{2\sqrt{2}}{3}$

And there you have it! We've successfully found the cosine of the angle. Wasn't that a thrilling trigonometric adventure?

Conclusion

In this problem, we've demonstrated how to find the cosine of an angle when we know its sine and the quadrant in which it lies. The unit circle, the Pythagorean identity, and the ASTC rule are the key tools in our trigonometric toolbox. Remember, the unit circle connects angles to coordinates, the Pythagorean identity relates sine and cosine, and ASTC helps us determine the signs of trigonometric functions in different quadrants. By mastering these concepts, you'll be well-equipped to tackle a wide range of trigonometry problems.

So, next time you encounter a problem like this, don't fret! Just remember the unit circle, the Pythagorean identity, ASTC, and the power is yours!