Question

Use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \cos \left(-\frac{\pi}{4}\right) $$

Short answer

\text{cos} \left(-\frac{\π}{4}\right) = \frac{\√2}{2}\.

Step-by-step solution

Identify the Function and Angle

We need to find the value of \(\cos \left(-\frac{\pi}{4}\right)\). This is a cosine function with an angle of \(-\frac{\pi}{4}\).

Recall the Even-Odd Properties of Cosine

Remember that cosine is an even function. This means \(\cos(-x) = \cos(x)\).

Apply the Even-Odd Property

Since cosine is even, we can write \(\cos \left(-\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right)\).

Evaluate the Cosine Value

Knowing that cosine of \(\frac{\text{π}}{4}\) is \(\frac{\text{√2}}{2}\), we find \(\text{cos} \left(\frac{\text{π}}{4}\right) = \frac{\text{√2}}{2}\).

Key concepts

Cosine Function

The cosine function, commonly written as \(\text{cos}(x)\), is one of the fundamental trigonometric functions. It relates the angle in a right-angled triangle to the ratio of the adjacent side over the hypotenuse. When dealing with the unit circle, the cosine of an angle corresponds to the x-coordinate of the point on the circle. Here are some key properties of the cosine function:

• It is periodic with a period of \(2\text{π} \) which means \(\text{cos}(x + 2\text{π}) = \text{cos}(x)\).
• It ranges from -1 to 1, inclusive.
• It is symmetric about the y-axis, meaning it is an even function.

Understanding these properties helps solve trigonometric problems without a calculator and is essential when evaluating angles outside common angles in small intervals.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved angles where both sides of the identity are defined. They are invaluable tools for simplifying and solving trigonometric problems. Some of the most useful trigonometric identities include:

• Pythagorean Identity: \(\text{sin}^2(x) + \text{cos}^2(x) = 1\)
• Angle Sum and Difference Identities: \(\text{cos}(a \pm b) = \text{cos}(a)\text{cos}(b) \mp \text{sin}(a)\text{sin}(b)\)
• Double Angle Identities: \(\text{cos}(2x) = 2\text{cos}^2(x) - 1\)
• Even-Odd Identities: \(\text{cos}(-x) = \text{cos}(x)\) and \(\text{sin}(-x) = -\text{sin}(x)\)

These identities simplify calculations and help us find the values of trigonometric functions for various angles efficiently.

Angle Evaluation

When dealing with angles in trigonometry, it's crucial to know how to evaluate them accurately, particularly when they are presented in special forms like radians. The problem given, \(\text{cos}\big(-\frac{\text{π}}{4}\big)\), demonstrates the process of evaluating an angle:

• Identify the function and angle: Here, it is the cosine function with an angle of \(-\frac{\text{π}}{4} \).
• Apply the even-odd property: Since cosine is an even function, \(\text{cos}(-x) = \text{cos}(x)\), simplifying evaluation for negative angles. Hence, \(\text{cos}\big(-\frac{\text{π}}{4}\big) = \text{cos}\big(\frac{\text{π}}{4}\big)\).
• Recall the cosine value: Knowing \(\text{cos}(\frac{\text{π}}{4}) = \frac{\text{√2}}{2}\) allows us to conclude that \(\text{cos}\big(-\frac{\text{π}}{4}\big) = \frac{\text{√2}}{2}\).

Accurate angle evaluation involves these steps: identifying the function and properties involved, and utilizing known values of trigonometric functions. This systematic approach ensures correct and efficient solutions.