Sec Θ Value Explained: Solve Tan² Θ = 3/8 Trig Problem

by Kenji Nakamura 55 views

Hey everyone! Today, we're diving into a trigonometric problem that might seem a bit tricky at first, but trust me, it's totally manageable once we break it down. We're given that $\tan^2 \theta = \frac{3}{8}$ and our mission, should we choose to accept it (and we do!), is to find the value of $\sec \theta$. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Core Trigonometric Relationship

First things first, let's recall a fundamental trigonometric identity that's going to be our best friend in solving this problem: the Pythagorean identity. This identity states that $\sin^2 \theta + \cos^2 \theta = 1$. Now, you might be wondering, "Okay, great, but how does this help us with $\tan^2 \theta$ and $\sec \theta$?" Patience, my friends, we're getting there! The key is to manipulate this identity to bring in our desired trigonometric functions. Remember that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sec \theta = \frac{1}{\cos \theta}$. Our goal is to somehow transform the Pythagorean identity to involve these ratios. To do this, we can divide the entire identity by $\cos^2 \theta$. This is a crucial step, so let's see how it unfolds:

sin⁡2θcos⁡2θ+cos⁡2θcos⁡2θ=1cos⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}

Simplifying this, we get:

tan⁡2θ+1=sec⁡2θ\tan^2 \theta + 1 = \sec^2 \theta

Boom! There it is. This is the golden equation that will lead us to the solution. It directly relates $\tan^2 \theta$ and $\sec^2 \theta$, which is exactly what we need. This powerful identity is derived directly from the basic Pythagorean identity and the definitions of tangent and secant. It's a cornerstone in trigonometry, and understanding how to derive and apply it can solve a lot of problems. So, remember this guys, this equation is your friend!

Applying the Given Information

Now that we have our magic equation, $\tan^2 \theta + 1 = \sec^2 \theta$, we can plug in the given value of $\tan^2 \theta$. We know that $\tan^2 \theta = \frac{3}{8}$, so let's substitute that in:

38+1=sec⁡2θ\frac{3}{8} + 1 = \sec^2 \theta

To add these together, we need a common denominator. We can rewrite 1 as $\frac{8}{8}$, giving us:

38+88=sec⁡2θ\frac{3}{8} + \frac{8}{8} = \sec^2 \theta

118=sec⁡2θ\frac{11}{8} = \sec^2 \theta

We're almost there! We now have the value of $\sec^2 \theta$, but we want the value of $\sec \theta$. To get that, we simply take the square root of both sides. But remember, when taking the square root, we need to consider both the positive and negative solutions. This is a critical step that's easy to overlook, but super important. Why? Because squaring either a positive or a negative number results in a positive number. So, the square root of a positive number can be either positive or negative.

Finding the Value of Sec θ

Taking the square root of both sides of $\frac{11}{8} = \sec^2 \theta$, we get:

sec⁡θ=±118\sec \theta = \pm \sqrt{\frac{11}{8}}

And there we have it! The value of $\sec \theta$ is $\pm \sqrt{\frac{11}{8}}$. This means that $\sec \theta$ can be either the positive square root of $\frac{11}{8}$ or the negative square root of $\frac{11}{8}$. The "$\pm$ " symbol is crucial here, because it acknowledges both possibilities. Without it, we'd only have half the picture! Remember that trigonometric functions can have both positive and negative values depending on the quadrant in which the angle $ heta$ lies.

Analyzing the Answer Choices

Now, let's take a look at the answer choices provided and see which one matches our solution:

A. $\pm \sqrt{\frac{8}{3}}$ B. $\pm \sqrt{\frac{11}{8}}$ C. $ rac{11}{8}$ D. $\frac{8}{3}$

We can clearly see that option B, $\pm \sqrt{\frac{11}{8}}$, is the correct answer. It perfectly matches our calculated value of $\sec \theta$. Options A, C, and D are incorrect. Option A has the fraction flipped, and options C and D only provide positive values, neglecting the negative possibility.

Key Takeaways

So, what have we learned today, guys? We've successfully tackled a trigonometric problem by:

  1. Recalling and understanding the fundamental Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. This is the bedrock of many trigonometric relationships, so make sure you have it down cold!
  2. Manipulating the Pythagorean identity to derive a crucial relationship between $\tan^2 \theta$ and $\sec^2 \theta$: $\tan^2 \theta + 1 = \sec^2 \theta$. This is a powerful tool in your trigonometric arsenal.
  3. Substituting the given value of $\tan^2 \theta$ into the derived equation.
  4. Taking the square root to solve for $\sec \theta$, remembering to consider both positive and negative solutions. This is a super important step to remember!
  5. Matching our solution to the correct answer choice.

This problem highlights the importance of not only knowing the trigonometric identities but also understanding how to manipulate them to solve for different variables. And remember, always consider both positive and negative solutions when taking square roots! Trigonometry can seem daunting, but by breaking it down step by step and understanding the underlying principles, you can conquer any problem that comes your way. Keep practicing, keep exploring, and keep having fun with math!

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By following these steps and understanding the core trigonometric relationships, you can confidently solve problems involving tangent and secant. Keep practicing and exploring the world of trigonometry!