Cos 50 Degree Sin 40 Degree + Sin 50 Degree Cos 40 Degree

Cos 50 Degree Sin 40 Degree + Sin 50 Degree Cos 40 Degree

The correct answer and explanation is :

The given expression is: cos⁡50∘sin⁡40∘+sin⁡50∘cos⁡40∘\cos 50^\circ \sin 40^\circ + \sin 50^\circ \cos 40^\circ

Step 1: Use the Angle Sum Identity

We recognize that this expression matches the sine angle sum identity: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B

Comparing, we see:

• A=50∘A = 50^\circ
• B=40∘B = 40^\circ

Applying the identity: sin⁡(50∘+40∘)=sin⁡90∘\sin(50^\circ + 40^\circ) = \sin 90^\circ

Step 2: Calculate the Value of sin⁡90∘\sin 90^\circ

We know that: sin⁡90∘=1\sin 90^\circ = 1

Final Answer:

cos⁡50∘sin⁡40∘+sin⁡50∘cos⁡40∘=1\cos 50^\circ \sin 40^\circ + \sin 50^\circ \cos 40^\circ = 1

Explanation (300 Words)

Trigonometric identities are powerful tools for simplifying expressions. Here, we used the sine angle sum identity, which states: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B

This formula is derived from the unit circle and the properties of right-angled triangles. It allows us to break down or combine sine and cosine terms.

For this specific problem, we noticed that the given expression had the exact form of the sine addition identity. By substituting A=50∘A = 50^\circ and B=40∘B = 40^\circ, we rewrote the expression as sin⁡(50∘+40∘)\sin(50^\circ + 40^\circ), which simplifies directly to sin⁡90∘\sin 90^\circ.

Since the sine of 90 degrees is always equal to 1, the final result is simply: 11

This method is a great example of how recognizing trigonometric identities can simplify seemingly complex expressions. Instead of manually calculating each sine and cosine term separately, we used the identity to quickly find the answer. This approach is commonly used in physics, engineering, and mathematics to solve problems efficiently.

Thus, the given expression simplifies to 1 using the sine angle sum identity.