Problem 55 - Algebra and Trigonometry

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. I expressed \(\cos 47^{\circ}+\cos 59^{\circ}\) as \(2 \cos 53^{\circ} \cos 6^{\circ}\)

Step-by-Step Solution

Answer

Yes, the statement makes sense. The given expression \(\cos 47^{\circ}+\cos 59^{\circ}\) can indeed be expressed as \(2 \cos 53^{\circ} \cos 6^{\circ}\) due to the sum-to-product identities in trigonometry.

Step 1: Write the formula

For any two angles A and B, we have the following trigonometric identity: \(cos A + cos B = 2 cos\frac{A + B}{2} cos\frac{A - B}{2}\)

Step 2: Substitute your given values into the identity

Substitute \( A = 47^{\circ}\) and \( B = 59^{\circ}\) into the identity: \[2 cos\frac{47^{\circ} + 59^{\circ}}{2} cos\frac{47^{\circ} - 59^{\circ}}{2}\]

Step 3: Simplify the expression

Simplify the fractions in the expression: \[2 cos 53^{\circ} cos -6^{\circ}\] Now, cosine is an even function, \(cos(-x) = cos(x)\), so we rewrite the expression as: \[2 cos 53^{\circ} cos 6^{\circ}\]

Key Concepts

Cosine

The cosine function is one of the primary trigonometric functions in mathematics. It is usually abbreviated as 'cos' and relates the angle in a right-angled triangle to the ratio of the length of the adjacent side to the hypotenuse.
The cosine of an angle in a right triangle is given by the formula:

• Cosine of angle = Adjacent side / Hypotenuse

It is essential in many areas such as mathematics, physics, and engineering, due to its involvement in wave motion and oscillations.
For any angle, the cosine function is periodic, meaning it repeats after a set interval—in this case, every 360 degrees or 2π radians. This periodicity is crucial for solving problems involving cycles.
In this exercise, we used the property of cosine in combination with trigonometric identities to simplify an expression. This involved adding two cosine terms using a specific identity, showcasing how adaptable the cosine function can be.

Simplifying Expressions

Simplifying trigonometric expressions can make calculations easier and solutions more apparent.
In trigonometry, simplifying often involves using identities that transform more complicated expressions into simpler forms.
An important trigonometric identity used in simplification is:

• For any two angles, A and B:
  \[ \cos A + \cos B = 2 \cos\frac{A + B}{2} \cos\frac{A - B}{2} \]

This particular identity can be useful when you are given the sum of two cosine terms and need to express it in terms of a product.
In our initial problem, the expression \( \cos 47^\circ + \cos 59^\circ \) was simplified using this identity, converting it into \( 2 \cos 53^\circ \cos 6^\circ \). This simplification makes further calculations or interpretations more straightforward.

Even Functions

An even function is one where the function value for a negative argument is equal to the function value for the positive argument. This property is fundamental in understanding how certain functions behave and apply under transformations.
Mathematically, a function \( f \) is even if:

• \( f(-x) = f(x) \)

The cosine function is a well-known example of an even function. This property is visible when you graph the function, as cosine demonstrates symmetry about the y-axis.
In the problem given, we used the even function property of cosine to simplify the expression \( \cos(-6^{\circ}) \) to \( \cos(6^{\circ}) \). This allows us to carry out computations without regard to sign, simplifying the expression handling substantially.
Understanding even functions is crucial in various mathematics fields, as this property often guides function manipulation, integration, and solving equations.