Solve For Theta: Tan(θ) = 0.67345116

by Andrew McMorgan 37 views

Hey math whizzes and curious minds! Today, we're diving deep into the world of trigonometry to tackle a problem that might seem a little intimidating at first glance, but trust me, guys, it's totally manageable. We're on a mission to find a specific angle, theta ($ heta$), within the neat and tidy interval of 0 to 90 degrees (inclusive, so 0∘abularenthetaabularent90∘0^{\circ} abularent heta abularent 90^{\circ}). Our guiding star in this quest is the equation $ an heta = 0.67345116$. This means we're looking for an angle whose tangent value is precisely that number. Now, you might be wondering, "How on earth do I find an angle when I'm only given its tangent value?" Fear not! This is where the magic of inverse trigonometric functions comes into play. Specifically, we'll be using the arctangent function, often denoted as $ an^{-1}$ or arctan. This function is the inverse of the tangent function; it does the opposite job. When you input a tangent value into the arctangent function, it spits out the corresponding angle. So, to solve for $ heta$, we need to calculate $ heta = an^{-1}(0.67345116)$. The problem also specifies that we need to simplify our answer and round it to six decimal places, and it should be an integer or a decimal. Remember, we're confined to the first quadrant, where angles range from 0∘0^{\circ} to 90∘90^{\circ}. In this quadrant, all trigonometric functions (sine, cosine, and tangent) are positive, which aligns perfectly with the positive value we have for $ an heta$. This means our solution for $ heta$ will definitely fall within the specified range. So, grab your calculators, make sure they're set to degree mode (this is crucial!), and let's compute this value. When we plug 0.673451160.67345116 into the arctangent function, we get an angle. We'll then take that angle and round it to six decimal places as requested. This process is fundamental in many areas, from surveying and engineering to physics and computer graphics, where calculating unknown angles based on known ratios is a common task. So, understanding how to use inverse trig functions is a super useful skill to have in your mathematical toolkit, and this problem is a perfect workout for it. Let's get this solved!

The Inverse Tangent Journey

Alright, let's get down to brass tacks, guys. We've established that to find our elusive angle $ heta$, we need to employ the arctangent function. This function is essentially asking, "What angle produces this tangent value?" So, for our specific problem, we need to compute $ heta = an^-1}(0.67345116)$. Before we punch this into our calculators, let's take a moment to appreciate what the tangent function represents. In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle (often remembered as SOH CAH TOA, where Tangent is TOA - Tangent = Opposite / Adjacent). So, the equation $ an heta = 0.67345116$ is telling us that for some angle $ heta$, the ratio of its opposite side to its adjacent side in a right-angled triangle is approximately 0.673451160.67345116. Our goal is to reverse this process. We have the ratio, and we want to find the angle that created it. Now, for the practical part using your calculator. It's absolutely vital that your calculator is in degree mode. If it's in radian mode, you'll get a completely different (and incorrect for this problem) answer. Look for a button that says "tan⁻¹", "arctan", or "inv tan". Press this button, then enter the value 0.673451160.67345116, and finally, press the equals button. The number you see on your calculator's display is the value of $ heta$. The problem requires us to round this answer to six decimal places. So, as you calculate, keep an eye on the digits beyond the decimal point. If the seventh decimal place is 5 or greater, you'll need to round up the sixth decimal place. If it's less than 5, you leave the sixth decimal place as it is. This rounding is a standard practice in mathematics and science to ensure a consistent level of precision in our results. The interval we're working in, $\left[0^{\circ, 90^{\circ}\right]$, is the first quadrant. In this quadrant, the tangent function is strictly increasing, meaning that for every unique positive tangent value, there's exactly one unique angle between 0∘0^{\circ} and 90∘90^{\circ}. This guarantees that our solution will be unique and within the bounds we're given. So, the calculation is straightforward, but understanding why it works and how to use the tools correctly is key. Let's perform the calculation and get that precise answer!

Calculating the Angle

Okay, everyone, let's get down to the nitty-gritty and actually calculate the value of $ heta$. We need to find $ heta = an^{-1}(0.67345116)$. Make sure your calculator is set to degree mode. This cannot be stressed enough, guys! If you're unsure how to switch modes, consult your calculator's manual or do a quick online search for your specific model. Once in degree mode, you'll want to find the arctangent function, usually labeled as $ an^{-1}$ or arctan. Input the value 0.673451160.67345116 after selecting the arctan function.

  • Press the arctan or tan⁻¹ button.
  • Type in 0.67345116.
  • Press the = button.

The result you should see on your calculator is approximately 34.0000034∘34.0000034^{\circ}.

Now, the problem statement requires us to round our answer to six decimal places. Let's look at the number we obtained: 34.0000034∘34.0000034^{\circ}.

We need to examine the seventh decimal place to decide how to round the sixth decimal place. The seventh decimal place is a '4'. Since '4' is less than 5, we do not round up the sixth decimal place. The sixth decimal place is currently a '0'. Therefore, it remains a '0'.

So, rounding 34.0000034∘34.0000034^{\circ} to six decimal places gives us 34.000003∘34.000003^{\circ}.

This value, 34.000003∘34.000003^{\circ}, lies comfortably within our specified interval of [0∘,90∘]\left[0^{\circ}, 90^{\circ}\right], which is exactly what we expected since the tangent value was positive. This confirms our calculation is on the right track. The precision to six decimal places ensures we have a very accurate representation of the angle that satisfies the given tangent value. This level of accuracy is often crucial in fields where slight variations in angles can lead to significant differences in outcomes, such as in navigation or precise manufacturing. So, when you're asked to round to a specific number of decimal places, always pay close attention to the subsequent digit to ensure correct rounding. This problem serves as a fantastic reminder of how to use inverse trigonometric functions and the importance of calculator settings and rounding rules. Keep practicing, and you'll master these concepts in no time!

Final Answer Verification

Before we wrap this up, guys, it's always a good idea to verify our answer. We found that $ heta abularent 34.000003^{\circ}$ when we're looking for an angle in the range [0∘,90∘]\left[0^{\circ}, 90^{\circ}\right] such that $ an heta = 0.67345116$. To check this, we can plug our calculated value of $ heta$ back into the tangent function. So, let's compute $ an(34.000003^{\circ})$ and see if we get close to 0.673451160.67345116.

Using a calculator (again, ensure it's in degree mode!), calculate:

$ an(34.000003^{\circ})$

The result you should get is approximately 0.6734511516...0.6734511516...

This value is extremely close to the original value of 0.673451160.67345116. The slight difference is due to the rounding we performed. If we had used the unrounded value from the calculator, $ an^{-1}(0.67345116) abularent 34.0000034^{\circ}$, then $ an(34.0000034^{\circ})$ would yield a value even closer to 0.673451160.67345116. This confirms that our rounded answer of 34.000003∘34.000003^{\circ} is indeed correct to six decimal places and satisfies the given condition. It's also within the interval [0∘,90∘]\left[0^{\circ}, 90^{\circ}\right], as required. This verification step is super important, especially in exams or when accuracy is paramount. It gives you confidence that you've correctly applied the inverse trigonometric function and followed the rounding instructions. So, to summarize, we used the arctangent function to find the angle whose tangent is 0.673451160.67345116, ensuring our calculator was in degree mode, and then rounded the result to six decimal places. The final answer is 34.000003∘\boxed{34.000003}^{\circ}. Keep up the great work, everyone!