Problem 1 - Algebra and Trigonometry

Question

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) Find the exact value of each expression. $$ \cos \left(45^{\circ}-30^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value of \(\cos(45° - 30°)\) is \(\sqrt{6}/4 + \sqrt{2}/4\).

Step 1: Identify the Given Values

The exercise provides two angles: a = 45°, and b = 30°.

Step 2: Use the Appropriate Formula

The formula for the cosine of the difference of two angles is: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).

Step 3: Substitute the Given Values into the Formula

Substitute a with 45° and b with 30° in the formula: \(\cos(45° - 30°) = \cos 45° \cos 30° + \sin 45° \sin 30°\).

Step 4: Find the Cosine and Sine Values

Look up or calculate these values using a unit circle or trigonometric table: \(\cos 45° = \sqrt{2}/2, \cos 30° = \sqrt{3}/2, \sin 45° = \sqrt{2}/2, \sin 30° = 1/2\). Substitute these into the equation.

Step 5: Substitute and Simplify

Substitute these values back into the formula and simplify: \(\sqrt{2}/2 * \sqrt{3}/2 + \sqrt{2}/2 * 1/2 = \sqrt{6}/4 + \sqrt{2}/4\). Adding these values gives the final result.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any angle. These identities are essential tools in trigonometry, helping us to simplify expressions and solve problems efficiently. One of the most critical identities is the cosine of the difference of two angles, given by: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] This formula is particularly useful when evaluating the cosine of an angle that is not straightforward, such as a combination of two known angles. By applying this identity, you can break down complex problems into simpler, manageable parts.

• The expression \(\cos(45° - 30°)\) can be evaluated using this identity.
• This formula combines the cosine and sine of individual angles to find the final cosine value of their difference.

Understanding trigonometric identities can greatly enhance your problem-solving skills, making complex calculations more accessible.

Exact Trigonometric Values

Exact trigonometric values are specific values for the trigonometric functions, like sine and cosine, that can be determined without a calculator. These are particularly useful when working with common angles found in the unit circle, such as 30°, 45°, and 60°. Memorizing these values can speed up problem-solving and improve your understanding of trigonometry. For example:

• \(\cos 45° = \sqrt{2}/2\)
• \(\cos 30° = \sqrt{3}/2\)
• \(\sin 45° = \sqrt{2}/2\)
• \(\sin 30° = 1/2\)

These exact values come from the properties of special triangles and the unit circle. In our original problem, we use these values to substitute into the cosine difference formula. This substitution is a crucial step in finding the exact trigonometric value of the expression \(\cos(45° - 30°)\). Knowing these values saves time and ensures accuracy in solving trigonometric equations.

Simplifying Expressions

Simplifying trigonometric expressions involves combining like terms and applying identities to make expressions easier to understand and solve. In the case of \(\cos(45° - 30°)\), after substituting the exact values into the identity formula, the next step is simplification. By substituting \(\cos 45°\), \(\cos 30°\), \(\sin 45°\), and \(\sin 30°\) into: \[ \cos(45° - 30°) = \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} * \frac{1}{2} \] The equation simplifies to: \[ \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \] Combine the fractions to get the final result: \[ \frac{\sqrt{6} + \sqrt{2}}{4} \] Simplification is about making the expression as concise as possible while maintaining the exactness of the answer. This process not only helps in reducing computational errors but also highlights the relationships between trigonometric functions.