Sin10°+sin20°+sin30°/cos10°+cos20°+cos30°=tan (?°) solve the question

To solve the equation \(\frac{\sin 10° + \sin 20° + \sin 30°}{\cos 10° + \cos 20° + \cos 30°} = \tan(?°)\), we will break it down step by step. The goal is to express the left side in a form that can help us identify the angle on the right side of the equation.

Step 1: Calculate the Sine Values

First, let’s find the sine values of the angles involved:

• \(\sin 30° = \frac{1}{2}\)
• \(\sin 20°\) and \(\sin 10°\) can be approximated or calculated using a calculator, but we will keep them in their sine form for now.

Step 2: Calculate the Cosine Values

Next, we need the cosine values:

• \(\cos 30° = \frac{\sqrt{3}}{2}\)
• \(\cos 20°\) and \(\cos 10°\) will also be kept in their original form for simplicity.

Step 3: Substitute Values into the Equation

Now we can substitute the known values into the expression:

\(\sin 10° + \sin 20° + \frac{1}{2}\) and \(\cos 10° + \cos 20° + \frac{\sqrt{3}}{2}\).

Step 4: Use Trigonometric Identities

To simplify the expression, we can use the sum of angles identities or the sine and cosine addition formulas, but for this case, let’s focus on combining the sine and cosine values.

Step 5: Combine and Simplify

We will calculate the numerator and denominator separately:

• Numerator: \(\sin 10° + \sin 20° + \frac{1}{2}\)
• Denominator: \(\cos 10° + \cos 20° + \frac{\sqrt{3}}{2}\)

Step 6: Find a Common Angle

Instead of calculating each term, we can look for patterns. Notice that \(\sin 30°\) and \(\cos 30°\) are complementary, which might help in finding a common angle.

Step 7: Use the Tangent Function

The left-hand side is structured as a tangent function, where we might express it as \(\frac{\sin A}{\cos A} = \tan A\). If we compute the simplified values, we can find an angle \(A\) such that:

\(\tan A\) corresponds to the value of the left-hand side. After some calculations, you will find that:

\(\frac{\sin 10° + \sin 20° + \sin 30°}{\cos 10° + \cos 20° + \cos 30°} \approx \tan 20°\).

Final Result

Thus, the angle that satisfies the equation is:

? = 20°

This result can be verified by inputting the values into a calculator or using a trigonometric table. Always check your calculations to ensure accuracy!