Hey guys! Let's dive into a trigonometry problem where we're given the value of sin θ and need to figure out tan θ. This is a classic problem that combines the definitions of trigonometric ratios and the Pythagorean theorem. So, buckle up, and let's get started!
Understanding the Problem
We're given that sin θ = 5/13, and our mission is to find tan θ. To do this, we'll need to recall the definitions of these trigonometric functions in terms of the sides of a right-angled triangle. Remember, SOH CAH TOA is your best friend here!
- Sin θ = Opposite / Hypotenuse
- Cos θ = Adjacent / Hypotenuse
- Tan θ = Opposite / Adjacent
Visualizing the Triangle
Think of a right-angled triangle where θ is one of the acute angles. Since sin θ = 5/13, we can say that the side opposite to θ is 5 units, and the hypotenuse (the longest side) is 13 units. Now, we need to find the length of the adjacent side to calculate tan θ.
Using the Pythagorean Theorem
The Pythagorean theorem comes to our rescue! It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as:
a² + b² = c²
Where:
- a and b are the lengths of the two shorter sides (legs) of the triangle.
- c is the length of the hypotenuse.
In our case:
- Opposite side (a) = 5
- Adjacent side (b) = ? (This is what we need to find)
- Hypotenuse (c) = 13
Plugging these values into the Pythagorean theorem, we get:
5² + b² = 13²
25 + b² = 169
Solving for the Adjacent Side
Now, let's solve for b:
b² = 169 - 25
b² = 144
Taking the square root of both sides:
b = √144
b = 12
So, the length of the adjacent side is 12 units. Great! We're one step closer to finding tan θ.
Calculating tan θ
Now that we know the lengths of the opposite and adjacent sides, we can easily calculate tan θ using its definition:
tan θ = Opposite / Adjacent
tan θ = 5 / 12
And there you have it! tan θ = 5/12.
Choosing the Correct Option
Looking at the options provided:
A. 13/12 B. 5/12 C. 12/13 D. 12/5
The correct answer is B. 5/12.
Wrapping Up
So, guys, we successfully found tan θ by using the given value of sin θ, visualizing a right-angled triangle, applying the Pythagorean theorem, and finally, using the definition of tan θ. Remember, these trigonometric ratios and the Pythagorean theorem are fundamental tools in trigonometry, and mastering them will help you tackle a wide range of problems. Keep practicing, and you'll become a trigonometry whiz in no time! If you have any questions, feel free to ask. Let's keep learning and growing together! This problem illustrates a common type of question in trigonometry, where you're given one trigonometric ratio and asked to find another. The key is to use the definitions of the ratios and the Pythagorean theorem to relate the sides of the right triangle. By understanding these concepts, you can confidently solve similar problems. Remember to always double-check your work and make sure your answer makes sense in the context of the problem.
Common Mistakes to Avoid
When solving problems like this, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer:
- Incorrectly Applying the Pythagorean Theorem: Make sure you have the correct sides in the equation a² + b² = c². Remember that c is always the hypotenuse, the longest side of the right triangle.
- Mixing Up Trigonometric Ratios: It's easy to mix up the definitions of sine, cosine, and tangent. Always double-check SOH CAH TOA to make sure you're using the correct ratio.
- Forgetting to Simplify: After finding the sides of the triangle, make sure you simplify the trigonometric ratio to its simplest form. For example, if you get tan θ = 10/24, simplify it to 5/12.
- Not Drawing a Diagram: Drawing a diagram can be incredibly helpful in visualizing the problem and ensuring you have the correct sides labeled. It can also help you avoid mistakes in applying the Pythagorean theorem and trigonometric ratios.
By keeping these common mistakes in mind, you can improve your accuracy and problem-solving skills in trigonometry. Practice makes perfect, so keep working on these types of problems, and you'll become more confident in your abilities.
Further Practice
To solidify your understanding of this concept, try solving similar problems with different given values. For example:
- If cos θ = 8/17, find sin θ and tan θ.
- If tan θ = 3/4, find sin θ and cos θ.
- If sin θ = 1/2, find cos θ and tan θ.
Working through these problems will give you more practice in applying the definitions of trigonometric ratios and the Pythagorean theorem. You can also explore more complex problems that involve multiple steps or require you to use trigonometric identities. The more you practice, the better you'll become at solving these types of problems.
Remember, trigonometry is a building-block subject, so mastering the fundamentals is crucial for success in more advanced topics. Keep practicing, and don't hesitate to seek help when you need it. With persistence and a solid understanding of the concepts, you'll excel in trigonometry!
Real-World Applications
Trigonometry isn't just a theoretical subject; it has numerous real-world applications in various fields. Understanding trigonometric ratios and their relationships can help you solve practical problems in fields like:
- Engineering: Engineers use trigonometry to calculate angles and distances in construction, surveying, and mechanical design.
- Navigation: Sailors and pilots use trigonometry to determine their position and course.
- Physics: Trigonometry is used to analyze motion, forces, and waves.
- Computer Graphics: Trigonometry is essential for creating 3D graphics and animations.
- Astronomy: Astronomers use trigonometry to measure the distances to stars and planets.
By learning trigonometry, you're not just mastering a mathematical concept; you're also gaining skills that can be applied in a wide range of real-world scenarios. So, keep exploring and discovering the fascinating applications of trigonometry in the world around you!