Calculate Tan(θ) With Terminal Ray Slope

Alright, guys, let's dive into a fun math problem where we need to figure out the tangent of an angle (that's tan θ) using some cool info about where the angle's terminal ray hits the unit circle. We're given that the slope of this terminal ray is -0.466, and our mission, should we choose to accept it, is to calculate tan θ rounded to three decimal places. Sounds like a plan? Let's break it down step by step so it's super clear and easy to follow.

Understanding the Problem

First off, let's get comfy with what we're dealing with. Imagine a circle – a unit circle, to be exact. This means its radius is exactly 1. Now, picture an angle θ starting from the positive x-axis and sweeping either counter-clockwise (for positive angles) or clockwise (for negative angles). The terminal ray is just the line that marks where the angle stops. This line intersects the unit circle at a point, and that point has coordinates (x, y). The slope of this terminal ray is given as -0.466. Remember, the slope of a line is rise over run, or in this case, the change in y divided by the change in x (Δy/Δx).

Why is this important? Well, in trigonometry, the tangent of an angle, tan θ, is defined as the ratio of the sine of the angle to the cosine of the angle (sin θ / cos θ). On the unit circle, the sine of the angle is the y-coordinate of the point where the terminal ray intersects the circle, and the cosine of the angle is the x-coordinate. So, tan θ = y/x. But hey, y/x is also the slope of the line! That's the key insight here.

Since we know the slope of the terminal ray, which is -0.466, and we know that the slope is equal to y/x, we can directly relate this to tan θ. Therefore, tan θ is simply the slope of the terminal ray. Easy peasy, right?

Key Concepts Recap:

• Unit Circle: A circle with a radius of 1 centered at the origin.
• Terminal Ray: The line that marks the end of an angle when drawn in standard position.
• Slope: Rise over run, or Δy/Δx.
• Tangent (tan θ): The ratio of the sine to cosine (sin θ / cos θ), which equals y/x on the unit circle.

Applying the Given Information

We're told that the slope of the terminal ray is -0.466. As we've established, the slope of the terminal ray is equivalent to tan θ. So, we can confidently say that:

tan θ = -0.466

But hold on! The question asks us to round the answer to three decimal places. Well, guess what? Our value is already given to three decimal places. How convenient is that?

Therefore, after rounding (which doesn't change anything in this case), we have:

tan θ = -0.466

And that's it! We've found our answer. Sometimes, math problems are simpler than they initially appear, especially when you understand the underlying concepts.

Let's Summarize the Steps:

1. Understand the Problem: Recognize the relationship between the terminal ray, its slope, and the tangent of the angle.
2. Identify Key Information: Note the given slope of the terminal ray (-0.466).
3. Apply the Concept: Use the fact that the slope of the terminal ray equals tan θ.
4. State the Result: Conclude that tan θ = -0.466.
5. Round (if Necessary): Ensure the answer is rounded to the specified number of decimal places (already done for us!).

Additional Insights and Considerations

While this problem was straightforward, let's think about some related scenarios and how we might tackle them.

Scenario 1: Finding the Angle θ

What if, instead of finding tan θ, we wanted to find the angle θ itself? We would use the inverse tangent function (arctan or tan⁻¹) to find the angle whose tangent is -0.466.

θ = arctan(-0.466)

Using a calculator, we would find that:

θ ≈ -24.97 degrees

Or, if we want the angle in radians:

θ ≈ -0.436 radians

Keep in mind that the arctan function typically gives you an angle in the range of -90° to +90° (or -π/2 to +π/2 radians). Since the tangent function has a period of 180° (or π radians), there are infinitely many angles that have the same tangent. To find other possible angles, you can add or subtract multiples of 180° (or π radians).

Scenario 2: Given Coordinates Instead of Slope

Suppose we weren't given the slope directly, but instead, we were given the coordinates of the point where the terminal ray intersects the unit circle, say (x, y). We could then calculate the slope (and thus tan θ) using the formula:

tan θ = y/x

For example, if the point of intersection was (0.9, -0.414), then:

tan θ = -0.414 / 0.9 ≈ -0.46

Scenario 3: Dealing with Different Quadrants

It's also important to consider the quadrant in which the terminal ray lies. The tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants. Knowing the quadrant helps you determine the correct angle when using the inverse tangent function.

In our original problem, since tan θ is negative (-0.466), the terminal ray must lie in either the second or fourth quadrant. The arctan function will give you an angle in the fourth quadrant (between -90° and 0°). If you need an angle in the second quadrant, you would add 180° (or π radians) to the result from the arctan function.

Common Mistakes to Avoid

• Forgetting the Unit Circle Relationship: The key to solving this problem is understanding that on the unit circle, the coordinates (x, y) of the intersection point directly correspond to cosine and sine, respectively. And that y/x is tan θ.
• Incorrectly Calculating Slope: Make sure you're using the correct formula for slope (rise over run).
• Not Rounding to the Correct Decimal Places: Always double-check the instructions to see how many decimal places are required.
• Ignoring Quadrant Information: Be mindful of the quadrant in which the terminal ray lies, as this can affect the sign of the trigonometric functions and the correct angle to use.

Conclusion

So, there you have it! Calculating tan θ when given the slope of the terminal ray is a straightforward process once you grasp the fundamental concepts of trigonometry and the unit circle. Remember, the slope of the terminal ray is equal to tan θ (which is y/x), and you're good to go. Always pay attention to details like rounding and quadrant information to ensure you get the correct answer. Keep practicing, and you'll become a tan θ master in no time! And remember, math can be fun – especially when you break it down into manageable steps. Keep exploring, keep learning, and keep rocking those math problems!

Final Answer: tan θ = -0.466