Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the product-to-sum formulas to write the product as a sum or difference. $$10 \cos 75^{\circ} \cos 15^{\circ}$$

Step-by-Step Solution

Answer

The simplified sum of the product \(10 \cos 75^{\circ} \cos 15^{\circ}\) is 2.5.

Step 1: Identifying the values of A and B

Start by identifying the values of A and B in the given product. In this case, \(A = 75^{\circ}\) and \(B = 15^{\circ}\).

Step 2: Apply the Product-to-Sum Formula

Next, apply the appropriate product-to-sum formula, which is \[2 \cos A \cos B = \cos(A-B) + \cos(A+B)\]. Replace A and B with the identified values to obtain \[10 \cos 75^{\circ} \cos 15^{\circ} = 5 [\cos(75^{\circ}-15^{\circ}) + \cos(75^{\circ}+15^{\circ})]\].

Step 3: Simplify the expressions

The next step is to simplify the expressions inside the cosine functions to obtain the resulting sum. This results in \[5 \cos 60^{\circ} + 5 \cos 90^{\circ}\].

Step 4: Compute the cosines

The last step is to compute the value of each cosine term. The cosine of 60 degrees is 0.5 and the cosine of 90 degrees is 0. This simplifies to \[5 * 0.5 + 5 * 0 = 2.5 + 0 = 2.5\].

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics. They simplify complex expressions and solve equations involving trigonometric functions.
These identities are equations that are true for any value of the variables involved.
Identities provide relationships between different trigonometric functions like sine, cosine, and tangent.
Key trigonometric identities include:

• Pythagorean identities, such as \( \sin^2\theta + \cos^2\theta = 1 \).
• Reciprocal identities, like \( \csc\theta = \frac{1}{\sin\theta} \).
• Co-function identities, for example, \( \sin(90^\circ - \theta) = \cos\theta \).

Understanding these identities helps in solving trigonometric equations and is widely used in calculus, physics, and engineering applications.

Cosine Function

The cosine function is a fundamental part of trigonometry. It describes the relationship between the angle of a right triangle and the ratio of the length of the adjacent side to the hypotenuse.
The cosine of an angle \(\theta\) in a right triangle can be calculated using the formula:\[\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\] Cosine function properties include:

• It is an even function, meaning \( \cos(-\theta) = \cos\theta \).
• Its values range between -1 and 1.
• It is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians.

Cosine functions are used in various applications such as calculating distances, waves, and oscillations, and are pivotal in solving problems related to angles and lengths.

Sum and Difference Formulas

Sum and difference formulas are powerful tools for simplifying and evaluating trigonometric expressions.
These formulas relate the sines and cosines of sums and differences of angles to the sines and cosines of the angles themselves.The sum formula for cosine is:\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]The difference formula for cosine is:\[\cos(A - B) = \cos A \cos B + \sin A \sin B\]These formulas help in converting products of trigonometric functions into sums, as is seen in the product-to-sum and sum-to-product identities.
By using these formulas, you can simplify complex expressions and solve trigonometric problems efficiently.
They are particularly useful when working with angles that are not standard angles like \(30^{\circ}\), \(45^{\circ}\), or \(60^{\circ}\).