Find the value of (i) Sin 22 degrees cos 38 degrees + cos 22 degrees + sin 38 degrees

Find the value of (i) Sin 22 degrees cos 38 degrees + cos 22 degrees + sin 38 degrees.

The correct answer and explanation is:

Correct Answer: sin⁡22∘cos⁡38∘+cos⁡22∘sin⁡38∘=sin⁡(22∘+38∘)=sin⁡60∘=32≈0.866\sin 22^\circ \cos 38^\circ + \cos 22^\circ \sin 38^\circ = \sin(22^\circ + 38^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866

Detailed Explanation (300 words):

To solve the expression sin⁡22∘cos⁡38∘+cos⁡22∘sin⁡38∘\sin 22^\circ \cos 38^\circ + \cos 22^\circ \sin 38^\circ, the most efficient way is by recognizing it as a known trigonometric identity. The expression fits the sine addition formula, which is: sin⁡Acos⁡B+cos⁡Asin⁡B=sin⁡(A+B)\sin A \cos B + \cos A \sin B = \sin(A + B)

In this case, let A=22∘A = 22^\circ and B=38∘B = 38^\circ. Applying the identity: sin⁡22∘cos⁡38∘+cos⁡22∘sin⁡38∘=sin⁡(22∘+38∘)=sin⁡60∘\sin 22^\circ \cos 38^\circ + \cos 22^\circ \sin 38^\circ = \sin(22^\circ + 38^\circ) = \sin 60^\circ

The sine of 60 degrees is a commonly known exact value: sin⁡60∘=32≈0.866\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866

This result shows that using trigonometric identities simplifies complex-looking expressions into much simpler forms. Instead of calculating each term individually using a calculator and then adding the results, recognizing patterns and identities saves time and improves accuracy.

Understanding trigonometric identities is important in many fields such as physics, engineering, and architecture. These identities help solve problems involving wave patterns, forces, and motion. In practical situations, such as signal processing or designing structures, such simplifications help in faster calculations and accurate predictions.

Memorizing fundamental identities like the sine and cosine addition formulas helps to approach mathematical problems more strategically. It transforms long computations into short, elegant solutions.

The identity used here is especially useful when dealing with the sum or difference of angles. Recognizing angle relationships and applying formulas efficiently boosts problem-solving skills in trigonometry and across mathematics in general.