Question

Verify the identity $$ 2 \sin A \sin B=\cos (A-B)-\cos (A+B) $$ with \(A=50^{\circ}\) and \(B=15^{\circ}\).

Short answer

Question: Verify if the trigonometric identity $$2 \sin A \sin B=\cos (A-B)-\cos (A+B)$$ holds true for the angles \(A=50^{\circ}\) and \(B=15^{\circ}\). Answer: The trigonometric identity holds true for \(A=50^{\circ}\) and \(B=15^{\circ}\), as both the left-hand side and.right-hand side of the equation yield approximately 0.802.

Step-by-step solution

Calculate values for both sides of the identity

We will start by calculating the value of both sides of the identity for \(A=50^{\circ}\) and \(B=15^{\circ}\). Left-hand side (LHS): $$ 2 \sin A \sin B = 2 \sin 50^{\circ} \sin 15^{\circ} $$ Right-hand side (RHS): $$ \cos (A-B)-\cos (A+B) = \cos (50^{\circ}-15^{\circ}) - \cos (50^{\circ}+15^{\circ}) $$

Compare the results

Now, let's compute the LHS and RHS values and compare them. LHS: $$ 2 \sin 50^{\circ} \sin 15^{\circ} \approx 0.802 $$ RHS: $$ \cos (50^{\circ}-15^{\circ}) - \cos (50^{\circ}+15^{\circ}) \approx \cos 35^{\circ} - \cos 65^{\circ} \approx 0.802 $$ Since the values of LHS and RHS are approximately equal (0.802), we can conclude that the given trigonometric identity holds true for \(A=50^{\circ}\) and \(B=15^{\circ}\).

Key concepts

Understanding the Sine Function in Trigonometry

In the realm of trigonometry, the sine function is one of the fundamental building blocks. It is defined for acute angles as the ratio of the length of the side of the triangle opposite the angle to the length of the hypotenuse. The sine function, represented by \(\sin(\theta)\), is prevalent in various mathematical contexts, particularly in the study of right-angled triangles. But beyond the scope of geometry, the sine function is crucial in modeling periodic phenomena, such as sound and light waves.

The sine function oscillates between -1 and 1, embodying the wave-like nature of the periodic systems it often represents. A practical understanding of the sine function allows students to solve problems involving angles and their associated ratios. For instance, in the given textbook exercise, the sine function aids in verifying the trigonometric identity \(2 \sin A \sin B\) for specific values of A and B. Empowering students to grasp and confidently apply the sine function is vital for their development in trigonometry and beyond.

The Role of the Cosine Function

Definition and Applications

The cosine function, symbolized by \(\cos(\theta)\), completes the duo of the most central trigonometric functions. Mirroring the sine function's geometric definition, the cosine equates to the ratio of the adjacent side to the hypotenuse in a right-angled triangle when considering acute angles.

In the broader scope of trigonometry, the cosine function is indispensable in the analysis of periodic motions and waves, just like its sine counterpart. Its value also oscillates between -1 and 1, encapsulating the essence of these cyclical patterns.

Usage in Trigonometric Identities

In regards to trigonometric identities, the cosine function frequently emerges. In the textbook solution at hand, the cosine appears within the expression \(\cos(A-B) - \cos(A+B)\). As students encounter more complex identities, a deep comprehension of the cosine function and its properties is essential. Indeed, such understanding is necessary for identifying and solving trigonometric equations, an integral component of higher mathematics and physics.

Trigonometry Problem Solving

Mastering trigonometry involves more than memorizing functions and formulas; it's about developing problem-solving skills to tackle a wide array of mathematical challenges. When faced with a trigonometry problem, students are encouraged to take a strategic approach: identify what's given, what's to be proven, and the relationships between trigonometric functions that might aid in the solution.

Following the step-by-step process, as demonstrated in the exercise's solution, elucidates how identities are methodically verified. Students must first calculate the value of both sides of the identity separately and then compare these values to establish their equivalence, which is the ultimate goal of such problems.

Effective problem solving in trigonometry depends heavily on a strong grasp of trigonometric identities. These identities are not merely mathematic curiosities; they are powerful tools that simplify complex problems, proving invaluable in fields ranging from engineering to economics. Hence, highlighting the application of these techniques in exercises nurtures a student's ability to navigate through the abstract and often daunting world of trigonometry.