$$ \frac{45}{50}=\frac{\sin\left(\alpha\right)}{\sin\left(\alpha\right)\cdot\cos\left(\alpha\right)+\sin\left(10\right)\cdot\cos\left(10\right)} $$

To solve the equation 

\[ \frac{45}{50} = \frac{\sin\left(\alpha\right)}{\sin\left(\alpha\right) \cdot \cos\left(\alpha\right) + \sin\left(10^\circ\right) \cdot \cos\left(10^\circ\right)} \]

we can follow these steps:

1. Simplify the fraction on the left-hand side:

\[ \frac{45}{50} = \frac{9}{10} \]

So the equation becomes:

\[ \frac{9}{10} = \frac{\sin(\alpha)}{\sin(\alpha) \cos(\alpha) + \sin(10^\circ) \cos(10^\circ)} \]

2. Let $ k = \sin(\alpha) \cos(\alpha) + \sin(10^\circ) \cos(10^\circ) $. Therefore, the equation becomes:

\[ \frac{9}{10} = \frac{\sin(\alpha)}{k} \]

3. Cross-multiply to solve for $ k $:

\[ 9k = 10 \sin(\alpha) \]

\[ k = \frac{10 \sin(\alpha)}{9} \]

4. Substitute $ k $ back into the equation:

\[ \sin(\alpha) \cos(\alpha) + \sin(10^\circ) \cos(10^\circ) = \frac{10 \sin(\alpha)}{9} \]

5. Recognize that $ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) $. Therefore:

\[ \sin(\alpha) \cos(\alpha) = \frac{1}{2} \sin(2\alpha) \]

\[ \sin(10^\circ) \cos(10^\circ) = \frac{1}{2} \sin(20^\circ) \]

So the equation becomes:

\[ \frac{1}{2} \sin(2\alpha) + \frac{1}{2} \sin(20^\circ) = \frac{10 \sin(\alpha)}{9} \]

6. Multiply through by 2 to clear the fractions:

\[ \sin(2\alpha) + \sin(20^\circ) = \frac{20 \sin(\alpha)}{9} \]

7. Rearrange to isolate terms involving $\alpha$:

\[ \sin(2\alpha) = \frac{20 \sin(\alpha)}{9} - \sin(20^\circ) \]

To solve for $\alpha$, one possible method is to use numerical methods or graphing techniques to find the roots of the equation, as it involves a transcendental equation which might not be straightforward to solve algebraically.

Do you want details or have any questions?

Here are 5 related questions for further exploration:
1. What are the values of $\sin(10^\circ)$ and $\cos(10^\circ)$?
2. How can we use trigonometric identities to simplify complex equations?
3. What numerical methods can be used to solve transcendental equations?
4. How can graphing techniques be applied to find the roots of trigonometric equations?
5. What are some common transcendental equations encountered in trigonometry?

**Tip:** When solving trigonometric equations, leveraging identities can simplify the process and reveal insights that might not be immediately obvious.

$$ \frac{45}{50} = \frac{\sin(\alpha)}{\sin(\alpha) \cdot \cos(\alpha) + \sin(10^\circ) \cdot \cos(10^\circ)} $$

Learn how to solve the trigonometric equation $$ \frac{45}{50} = \frac{\sin(\alpha)}{\sin(\alpha) \cdot \cos(\alpha) + \sin(10^\circ) \cdot \cos(10^\circ)} $$ using identities such as $$ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) $$ and approaches to transcendental equations. This problem is ideal for high school students studying Pre-Calculus or Calculus.

Solve Trigonometric Equation Involving Sine and Cosine: $$ \frac{45}{50} = \frac{\sin(\alpha)}{\sin(\alpha) \cos(\alpha) + \sin(10^\circ) \cos(10^\circ)} $$

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation