Proving the Identity: sin140 - sin100 -sin120 Using Sum-to-Product Formulas

In this article, we will use the sum-to-product formula to show that sin140° - sin100° -sin120°. The sum-to-product formula is a powerful tool in trigonometry that helps simplify and manipulate trigonometric expressions. We will walk through the proof step-by-step, applying the formula and verifying the result using both analytical and numeric methods.

Step 1: Identifying A and B

To apply the sum-to-product formula, we first need to identify the angles A and B. In this case:

A 140° B 100°

Step 2: Calculating the Sums and Differences

The next step is to calculate the sum (A B) and the difference (A - B) of these angles:

A B 140° 100° 240° A - B 140° - 100° 40°

Step 3: Applying the Sum-to-Product Formula

Using the sum-to-product formula:

$$sin A - sin B 2 cosleft(frac{A B}{2}right) sinleft(frac{A - B}{2}right)$$

We can write:

$$sin 140^circ - sin 100^circ 2 cosleft(frac{240^circ}{2}right) sinleft(frac{40^circ}{2}right)$$

Simplifying the arguments:

$$frac{240^circ}{2} 120^circ$$ $$frac{40^circ}{2} 20^circ$$

Substituting these values back into the formula:

$$sin 140^circ - sin 100^circ 2 cos 120^circ sin 20^circ$$

Step 4: Calculating cos120°

To finish the proof, we need to calculate cos120°:

$$cos 120^circ -frac{1}{2}$$

Step 5: Substituting the Value of cos120°

Substitute cos120° back into the equation:

$$sin 140^circ - sin 100^circ 2 left(-frac{1}{2}right) sin 20^circ$$

Which simplifies to:

$$sin 140^circ - sin 100^circ -sin 20^circ$$

Step 6: Verifying the Final Equality

Now, we need to check if -sin20° equals -sin120°:

From trigonometric properties, we know:

$$sin 120^circ sin 60^circ frac{sqrt{3}}{2}$$

Thus:

$$-sin 120^circ -frac{sqrt{3}}{2}$$

However, substituting the numeric values:

$$sin 140^circ 0.6427876$$ $$sin 100^circ 0.9848077$$ $$sin 140^circ - sin 100^circ 0.6427876 - 0.9848077 -0.3420201$$ $$sin 120^circ 0.866025$$

Clearly, -sin20° and -sin120° do not equal numeric values of sin140 - sin100 and -sin120 respectively. Therefore, the simplified form does not match the original values.

Conclusion

The identity sin140 - sin100 -sin120 cannot be proven by the sum-to-product formula as the resulting expression does not match the original numeric values. This indicates a discrepancy between the analytical and numeric approaches. Nonetheless, the sum-to-product formula provides a valuable method for simplifying trigonometric expressions and can be a useful tool in various mathematical applications.