Solving Trigonometric Functions: A Step-by-Step Guide

Oct 22, 2025 by Dimemap Team 54 views

Hey everyone! Today, we're diving into the world of trigonometric functions and figuring out how to find their values at specific points. Don't worry, it's not as scary as it sounds! We'll break down the process step-by-step, making it super easy to understand. We'll be looking at two examples, so grab your pencils and let's get started. Specifically, we are going to find the value of functions like $y = 2 ext{sin}\,(x - rac{\pi}{6}) + 1$ and $y = -\text{sin}\,(x + rac{\pi}{4})$ and calculate the value when we know what $x$ is. This is a common task in mathematics, so understanding the basics is super important. We'll be using the sine function, which is a fundamental concept in trigonometry. Ready to become trigonometry masters? Let's go! This article is designed to help you find the value of a function, specifically trigonometric functions, by substituting known values of $x$ and solving the equation.

Part A: Calculating $y = 2 \text{sin}\,(x - rac{\pi}{6}) + 1$ at $x = rac{4\pi}{3}$

Alright, let's tackle the first problem: $y = 2 \text{sin}\,(x - rac{\pi}{6}) + 1$ , where $x = rac{4\pi}{3}$ . Our goal here is to find the value of $y$ when we plug in the given value of $x$ . The key here is to carefully substitute the value of $x$ and then simplify the expression using the properties of the sine function. Trust me, it's easier than it looks! First, we need to replace every instance of $x$ in the equation with $rac{4\pi}{3}$ . This gives us $y = 2 \text{sin}\,( rac{4\pi}{3} - rac{\pi}{6}) + 1$ . See? No biggie so far! Now, the next step is to simplify the expression inside the sine function. We need to subtract the fractions, which means we need a common denominator. The least common denominator for 3 and 6 is 6, so we convert $rac{4\pi}{3}$ to $rac{8\pi}{6}$ . The equation now becomes $y = 2 \text{sin}\,( rac{8\pi}{6} - rac{\pi}{6}) + 1$ . Then, we subtract the fractions inside the sine function: $rac{8\pi}{6} - rac{\pi}{6} = rac{7\pi}{6}$ . Therefore, we now have $y = 2 \text{sin}\,( rac{7\pi}{6}) + 1$ . The next step involves finding the sine of $rac{7\pi}{6}$ . Recall that the sine function gives the y-coordinate on the unit circle. The angle $rac{7\pi}{6}$ is in the third quadrant, where the sine function is negative. The reference angle for $rac{7\pi}{6}$ is $rac{\pi}{6}$ , and we know that $\text{sin}\,( rac{\pi}{6}) = rac{1}{2}$ . Thus, $\text{sin}\,( rac{7\pi}{6}) = - rac{1}{2}$ . We substitute this value back into our equation: $y = 2(- rac{1}{2}) + 1$ . Now, we simplify: $2 * - rac{1}{2} = -1$ , giving us $y = -1 + 1$ . Finally, $-1 + 1 = 0$ . So, the value of the function $y = 2 \text{sin}\,(x - rac{\pi}{6}) + 1$ at $x = rac{4\pi}{3}$ is 0. Easy peasy, right?

This method is super useful for when you need to find the value of the function. Always remember the order of operations and properties of trigonometric functions to avoid errors.

Part B: Calculating $y = -\text{sin}\,(x + rac{\pi}{4})$ at $x = - rac{\pi}{2}$

Now, let's move on to the second example: $y = -\text{sin}\,(x + rac{\pi}{4})$ , where $x = - rac{\pi}{2}$ . The process is pretty much the same: substitute the given value of $x$ and simplify. First, we replace $x$ with $- rac{\pi}{2}$ in the equation: $y = -\text{sin}\,(- rac{\pi}{2} + rac{\pi}{4})$ . Now, let's simplify the expression inside the sine function. We need to add $- rac{\pi}{2}$ and $rac{\pi}{4}$ . Again, we need a common denominator, which is 4. So, we convert $- rac{\pi}{2}$ to $- rac{2\pi}{4}$ . The equation becomes $y = -\text{sin}\,(- rac{2\pi}{4} + rac{\pi}{4})$ . Add the fractions: $- rac{2\pi}{4} + rac{\pi}{4} = - rac{\pi}{4}$ . Now we have $y = -\text{sin}\,(- rac{\pi}{4})$ . Remember, the sine function is an odd function, meaning $\text{sin}\,(-θ) = -\text{sin}\,(θ)$ . So, $\text{sin}\,(- rac{\pi}{4}) = -\text{sin}\,( rac{\pi}{4})$ . We know that $\text{sin}\,( rac{\pi}{4}) = rac{\sqrt{2}}{2}$ . Therefore, $\text{sin}\,(- rac{\pi}{4}) = - rac{\sqrt{2}}{2}$ . Substituting this back into our equation, we get $y = -(- rac{\sqrt{2}}{2})$ . Finally, a negative times a negative is a positive, so $y = rac{\sqrt{2}}{2}$ . Therefore, the value of the function $y = -\text{sin}\,(x + rac{\pi}{4})$ at $x = - rac{\pi}{2}$ is $rac{\sqrt{2}}{2}$ .

This example emphasizes the importance of understanding the properties of trigonometric functions, such as the odd and even properties of trigonometric functions. Using them can simplify calculations. Now you know how to find the value of a function in different scenarios!

Key Takeaways and Tips

Alright, let's recap some key takeaways and tips to make sure you've got this down: First, always remember the order of operations (PEMDAS/BODMAS) when simplifying expressions. This is crucial for calculating the function value correctly. Second, know your unit circle and the values of sine, cosine, and tangent for common angles. The unit circle is your best friend when dealing with trigonometric functions. Third, understand the properties of trigonometric functions, such as the odd and even properties of sine, cosine, and tangent. These properties can significantly simplify your calculations. Always remember that $\text{sin}\,(-x) = -\text{sin}\,(x)$ (odd function), $\text{cos}\,(-x) = \text{cos}\,(x)$ (even function), and $\text{tan}\,(-x) = -\text{tan}\,(x)$ (odd function). Practice makes perfect. The more you practice, the more comfortable you'll become with these calculations. Try working through different examples to reinforce your understanding. Make sure you understand the concepts of function evaluation and substitution. These are core mathematical ideas. Finally, don't be afraid to ask for help! If you're struggling, don't hesitate to ask your teacher, classmates, or use online resources for help. Learning trigonometry is a journey, so be patient with yourself, and enjoy the process!

Further Practice

Want to become a trigonometry ninja? Here are some extra practice problems you can try:

Find the value of $y = 3\text{cos}\,(2x) - 1$ when $x = rac{\pi}{4}$ .
Calculate $y = \text{tan}\,(x - rac{\pi}{3})$ when $x = \pi$ .
Determine the value of $y = \text{sin}\,(x) + \text{cos}\,(x)$ when $x = 0$ .

Work through these problems on your own, and check your answers. This will really help solidify your understanding of how to find the value of a function. You got this, guys! And remember, understanding how to find the value of a function is a fundamental skill in math and will also help you if you study calculus, physics or engineering. Good luck, and happy calculating!