Problem 98 - Precalculus with Limits

Question

In Exercises 95 - 98, verify the identity. \( a \sin B\theta + b \cos B\theta = \sqrt{a^2 + b^2} \cos\left(B\theta - C\right) \), where \( C = \arctan(a/b) \) and \( b > 0 \)

Step-by-Step Solution

Answer

The given trigonometric identity, \( a \sin B\theta + b \cos B\theta = \sqrt{a^2 + b^2} \cos(B\theta - C) \) where \( C = \arctan(a/b) \) and \( b > 0 \), holds true.

Step 1: Identify given formula

The given formula to prove is \( a \sin B\theta + b \cos B\theta = \sqrt{a^2 + b^2} \cos(B\theta - C) \), where \( C = \arctan(a/b) \) and \( b > 0 \).

Step 2: Express left side in form of cosine

We know that \( \sin(x-y) = \sin x \cos y - \cos x \sin y \). So, we can express the left hand side (LHS) as a single cosine function. In the formula, let \( x = B\theta \) and \( y = C \).So, \( \sin(x-y) \) becomes \( a \sin B\theta \cos C + b \cos B\theta \cos C \). Considering \( \cos C = \frac{b}{\sqrt{a^2+b^2}} \) as we know \( C = \arctan(a/b) \), we can substitute C in the equation, the LHS becomes \( \sqrt{a^2+b^2} \cos(B\theta - \arctan(a/b)) \).

Step 3: Compare with right side

After expressing the left side in the form of cosine, it is now exactly the same as the Right Hand Side (RHS) of the equation, which is \( \sqrt{a^2+b^2} \cos(B\theta - C) \).So, this confirms that the given identity is verified.

Key Concepts

Cosine

The cosine function, denoted as \( \cos \), is one of the fundamental trigonometric functions. It relates an angle in a right triangle to the ratio of the length of the adjacent side over the hypotenuse. In the unit circle, cosine corresponds to the x-coordinate of a point on the circle.
For an angle \( \theta \), \( \cos(\theta) \) helps to determine horizontal components in trigonometric expressions. The versatility of cosine lies in its ability to express angles and rotations compactly, which is crucial in trigonometry.
The formula \( \cos(A - B) = \cos A \cos B + \sin A \sin B \) is particularly useful when expressing the difference between two angles as a single component, exemplified in our given identity. Remember, understanding cosine in this way makes analyzing wave and circular motion much easier.

Arctangent Function

The arctangent function, denoted as \( \arctan \), is the inverse of the tangent function. It calculates the angle whose tangent is a given number. In our verified identity, \( C = \arctan(a/b) \) determines the angle whose tangent value, \( \frac{a}{b} \), relates to sides of a right triangle.
This function is essential for finding angles when the ratios of the sides are known, especially in real-world applications like navigation and engineering.
To understand its value, consider that arctangent translates a tangent ratio back into a meaningful angle. Thus, adopting \( C = \arctan(a/b) \) helps us simplify our trigonometric identity from earlier, integrating it neatly into the cosine expression.

Sine

The sine function, denoted as \( \sin \), is another primary trigonometric function. It describes the ratio of the length of the opposite side over the hypotenuse in a right triangle. On the unit circle, sine equates to the y-coordinate of a corresponding point.
In our equation \( a \sin B\theta + b \cos B\theta \) equals something known involving cosine. By linking \( \sin(\theta) \) with its complementary trigonometric counterpart \( \cos(90^\circ - \theta) \), it’s possible to find common links between different waves and oscillations.
By expressing the sine component as cosine, we get unification in trig calculations, helping bridge identities and manage complex equations with ease.

Proof Verification

When we talk about proof verification in mathematics, we're discussing the process of validating that an identity or equation is true for all valid inputs. A proof must be logically consistent and cover all bases to ensure correctness.
In our provided identity problem, proof verification involves equating the left and right sides to show they're identical, using algebraic manipulation and known trigonometric identities.
To achieve this, recognize that transforming the left-hand expressions into terms of single trig functions, as done with cosine here, allows for straightforward comparisons. Consistency in approach and thoroughness in step follow-through guarantees accurate verification of complex identities. This process is crucial in deepening comprehension of trigonometric relationships and functions.