The equation sin70° × sin20° sin40° × sin30° is a fascinating trigonometric identity that can be explored through various trigonometric identities and properties. Let's break down the solution step by step, using degrees for clarity.

Understanding the Identity

First, we need to clarify the identity and ensure it is correct within the context of degrees:

sin70° × sin20° sin40° × sin30°

By revisiting the trigonometric properties, specifically using the product-to-sum identities, we can prove this identity.

Using Trigonometric Relations

Method 1: Using the Cosine Difference Formula

According to the cosine difference formula, we have:

cos(A - B) cosAcosB sinAsinB

Setting (A 70^circ) and (B 20^circ), we get:

cos(70^circ - 20^circ) cos70^circ cos20^circ sin70^circ sin20^circ

Since (A - B 50^circ), we have:

cos50^circ cos70^circ cos20^circ sin70^circ sin20^circ

Using the co-function identity (cos50^circ sin40^circ), we get:

sin40^circ sin70^circ cos20^circ sin70^circ sin20^circ

Factoring out (sin70^circ) on the right-hand side, we get:

sin40^circ sin70^circ (cos20^circ sin20^circ)

Using the co-function identity again, (sin70^circ cos20^circ), we get:

sin40^circ cos20^circ (cos20^circ sin20^circ)

Simplifying, we get:

sin40^circ cos20^circ cos20^circ cos20^circ sin20^circ

This is equivalent to:

sin40^circ 2 cos20^circ cos20^circ 2 cos20^circ sin20^circ / 2

Which simplifies to:

sin40^circ 2 cos20^circ cos20^circ / 2 2 cos20^circ sin20^circ / 2

This is:

sin40^circ cos20^circ cos20^circ cos20^circ sin20^circ

Which is:

sin40^circ cos20^circ (cos20^circ sin20^circ)

Finally, we can use the identity (sin30^circ 1/2) to simplify:

sin30^circ sin40^circ 1/2 sin40^circ 1/2 (cos20^circ (cos20^circ sin20^circ))

This confirms that:

sin70^circ sin20^circ sin40^circ sin30^circ

Conclusion

Thus, the trigonometric identity sin70° × sin20° sin40° × sin30° holds true, as proven using trigonometric identities and properties of co-functions. The solution shows that when working in degrees, the identity is valid, but care must be taken when using radians, as the relationship changes.

Key Takeaways

Trigonometric Identities: Using identities such as the cosine difference formula and co-function identities are essential in verifying such trigonometric relationships. Degree vs. Radians: It is important to consider whether the identity holds for degrees or radians, as it can vary between the two systems. Verification Steps: Breaking down complex trigonometric expressions into simpler components simplifies the verification process.

Frequently Asked Questions

Q: Why is sin70° × sin20° sin40° × sin30°? Ans: The identity is derived using trigonometric identities and properties of co-functions. The detailed derivation is provided in the article. Q: Does this identity hold for radians? Ans: No, the identity specifically holds for degrees. Using radians would result in a different relationship. Q: Can you provide a graphical representation to verify the identity? Ans: Graphical verification is not typically used for such trigonometric identities, but visualizing the unit circle and the angles involved can offer intuitive understanding.

References

Wikipedia: Trigonometric Identities Wikipedia: List of Trigonometric Identities