Using the Pythagorean Identity

Oct 13, 2025 by windailys.com 149 views

Introduction

In trigonometry, right triangles with acute angles are a fundamental concept. Given the relationships between the cosine and sine of these angles, we can solve for the value of x. In this article, we will explore how to use the given equations to find the value of x.

The Problem

We are given a right triangle with two acute angles, ${$ \theta$}$ and ${$ \phi$}$, such that:

$\cos \phi = x + 0.2$ $\sin \theta = 2x - 0.5$

Our goal is to find the value of x.

Using the Pythagorean Identity

To solve for x, we can use the Pythagorean identity, which states that:

$\sin^2 \theta + \cos^2 \phi = 1$

We can substitute the given equations into this identity:

$(2x - 0.5)^2 + (x + 0.2)^2 = 1$

Expanding and simplifying the equation, we get:

$4x^2 - 2x + 0.25 + x^2 + 0.4x + 0.04 = 1$

Combine like terms:

$5x^2 + 0.4x + 0.34 = 1$

Subtract 1 from both sides:

$5x^2 + 0.4x - 0.66 = 0$

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, a = 5, b = 0.4, and c = -0.66. Plugging these values into the formula, we get:

$x = \frac{-0.4 \pm \sqrt{(0.4)^2 - 4(5)(-0.66)}}{2(5)}$

Simplifying the expression under the square root, we get:

$x = \frac{-0.4 \pm \sqrt{0.16 + 13.2}}{10}$

$x = \frac{-0.4 \pm \sqrt{13.36}}{10}$

$x = \frac{-0.4 \pm 3.65}{10}$

Finding the Value of x

We have two possible solutions for x:

$x = \frac{-0.4 + 3.65}{10}$ $x = \frac{-0.4 - 3.65}{10}$

Simplifying the first solution, we get:

$x = \frac{3.25}{10}$ $x = 0.325$

Simplifying the second solution, we get:

$x = \frac{-4.05}{10}$ $x = -0.405$

Conclusion

We have solved for the value of x in a right triangle with acute angles. The two possible solutions are x = 0.325 and x = -0.405. However, since x is a measure of the length of a side, it must be a positive value. Therefore, the correct solution is x = 0.325.

Answer

$The final answer is $\boxed{0.325}$<br/> **A Right Triangle with Acute Angles: Q&A**$

Introduction

In our previous article, we explored how to solve for the value of x in a right triangle with acute angles. We used the given equations and the Pythagorean identity to find the value of x. In this article, we will answer some common questions related to this topic.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states:

$\sin^2 \theta + \cos^2 \phi = 1$

This identity is used to relate the sine and cosine of two angles in a right triangle.

Q: How do I use the Pythagorean identity to solve for x?

A: To use the Pythagorean identity to solve for x, you need to substitute the given equations into the identity and simplify the resulting equation. In this case, we substituted the given equations into the Pythagorean identity and got:

$5x^2 + 0.4x - 0.66 = 0$

We then solved this quadratic equation using the quadratic formula to find the value of x.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, a = 5, b = 0.4, and c = -0.66. We plugged these values into the formula and solved for x.

Q: Why do I get two possible solutions for x?

A: When solving a quadratic equation, you can get two possible solutions. In this case, we got two possible solutions for x:

$x = \frac{-0.4 + 3.65}{10}$ $x = \frac{-0.4 - 3.65}{10}$

However, since x is a measure of the length of a side, it must be a positive value. Therefore, the correct solution is x = 0.325.

Q: How do I know which solution is correct?

A: To determine which solution is correct, you need to consider the context of the problem. In this case, we were given a right triangle with acute angles, and x is a measure of the length of a side. Since x must be a positive value, the correct solution is x = 0.325.

Q: What are some common mistakes to avoid when solving for x?

A: Some common mistakes to avoid when solving for x include:

Not using the Pythagorean identity correctly
Not simplifying the resulting equation correctly
Not using the quadratic formula correctly
Not considering the context of the problem

By avoiding these common mistakes, you can ensure that you get the correct solution for x.

Conclusion

In this article, we answered some common questions related to solving for x in a right triangle with acute angles. We covered topics such as the Pythagorean identity, the quadratic formula, and common mistakes to avoid. By following these tips and avoiding common mistakes, you can ensure that you get the correct solution for x.