Problem 20 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \cos 75^{\circ}-\cos 15^{\circ} $$

Step-by-Step Solution

Answer

The product's exact value for the given expression \(\cos 75^{\circ}-\cos 15^{\circ}\) is \(-\frac{\sqrt{2}}{2}\)

Step 1: Rearranging the given expression

Rearranging \(\cos 75^{\circ}-\cos 15^{\circ}\) we get \(-\cos 15^{\circ} + \cos 75^{\circ}\). Notice this is similar to \(\cos(A - B)\) expression, if we let A = 75° and B = 15°.

Step 2: Applying the Cosine Difference Identity

We know that \(\cos(A - B) = \cos A \cos B + \sin A \sin B\), let's apply this on the rearranged expression. Then, \(-\cos 15^{\circ} + \cos 75^{\circ} = -2 \sin(\frac{A + B}{2}) \sin (\frac{A - B}{2})\).

Step 3: Substituting A and B

Substitute A = 75° and B = 15° into the above equation. We now obtain \(-2 \sin(\frac{75^{°} + 15^{°}}{2}) \sin (\frac{75^{°} - 15^{°}}{2}) = -2 \sin (45^{°}) \sin (30^{°})\).

Step 4: Find the product's exact value

Using the sine value equivalents, we substitute the values into the equation. Here, \(\sin 45^{°} = \frac{\sqrt{2}}{2}\) and \(\sin 30^{°} = \frac{1}{2}\). Hence, the exact value of the product is \(-2 * \frac{\sqrt{2}}{2} * \frac{1}{2} = -\frac{\sqrt{2}}{2}\).

Key Concepts

Cosine Difference Identity

The cosine difference identity is a useful tool in trigonometry that helps simplify expressions involving cosine functions of two angles. Whenever you see cosine subtracted from another cosine, it might signal the use of this identity. The formula for the cosine difference is: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] In the exercise, the expression \( \cos 75^{\circ}-\cos 15^{\circ} \) can be rearranged as \( -\cos 15^{\circ} + \cos 75^{\circ} \). By recognizing this form, we can apply the cosine difference identity. This helps transform the difference into a product of sines, which is often easier to interpret or further manipulate mathematically.

Sine Values

Sine values are fundamental in trigonometry and arise frequently in various formulas, like the ones used here. Certain angles have well-known sine values that we refer to as exact trigonometric values. For instance:

• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

Knowing these values allows us to substitute directly into trigonometric expressions. In the context of our exercise, substituting \( \sin 45^{\circ} \) and \( \sin 30^{\circ} \) led us to find the exact value of the expression \( -2 \sin 45^{\circ} \sin 30^{\circ} = -\frac{\sqrt{2}}{2} \). These values simplify calculations and lead to precise results.

Exact Trigonometric Values

Exact trigonometric values are key values you should memorize. They provide precise results for specific angles without needing a calculator. Common angles like 0°, 30°, 45°, 60°, and 90° have known values for sine, cosine, and tangent. These serve as building blocks for evaluating and simplifying trigonometric expressions. For the sine function:

• \( \sin 0^{\circ} = 0 \)
• \( \sin 30^{\circ} = \frac{1}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 90^{\circ} = 1 \)

Having these at your fingertips makes solving trigonometric equations much more manageable and ensures accuracy in your calculations.

Angle Sum and Difference

The angle sum and difference identities are powerful tools that simplify trigonometric expressions and solve equations involving angles expressed as sums or differences. Here is a quick guide to the basic identities:

• Angle Sum Identity for Cosine: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• Angle Difference Identity for Cosine: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
• Angle Sum Identity for Sine: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• Angle Difference Identity for Sine: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

For the problem at hand, we used the cosine difference identity to break down \( \cos 75^{\circ}-\cos 15^{\circ} \) effectively. Such identities reduce complex expressions to familiar forms enabling easier computation or understanding.