Problem 222 - Precalculus

Question

For the following exercises, use reference angles to evaluate the expression. If \(\cos t=\frac{1}{2}, \quad\) what is the \(\cos (-t) ?\)

Step-by-Step Solution

Answer

The value of \( \cos(-t) \) is \( \frac{1}{2} \).

Step 1: Identify the Cosine Function Property

Recall the even property of the cosine function: \( \cos(-t) = \cos(t) \). This means that the cosine of a negative angle is the same as the cosine of the positive angle.

Step 2: Apply Property to Given Expression

Using the even property from Step 1, substitute the given value of \( \cos(t) = \frac{1}{2} \) into the expression. Thus, \( \cos(-t) = \cos(t) = \frac{1}{2} \).

Step 3: Verify Using Reference Angle

Remember that the cosine function repeats every \( 360^\circ \) or \( 2\pi \) radians. Therefore, switching the sign of \( t \) to \( -t \) just mirrors the angle over the x-axis in the unit circle, not affecting the cosine value.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that helps us understand the relationship between the angles and sides of a triangle. It is particularly useful in dealing with right triangles. Here’s why it’s so important:

• Cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• The function has a period of \[2\pi\] radians or \[360^\circ\], meaning it repeats its values over these intervals.

Understanding the cosine function helps us solve problems related to angles, especially when combined with its properties and the unit circle. The unit circle model is particularly valuable as it links angle measures to coordinates in a circle with a radius of one.

Reference Angles

A reference angle is a positive acute angle that helps find the trigonometric function values of other angles. By using reference angles, we simplify complex problems – keeping them in the first quadrant. Here’s how this works:

• The reference angle is always measured from the x-axis.
• To calculate the reference angle, you take the absolute value of the given angle and reduce it to a value less than \[90^\circ\] or \[\frac{\pi}{2}\] radians.

Reference angles allow us to tap into known values like \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and apply them to other quadrants, helping in computing cosine or any other trigonometric function efficiently.

Even Properties of Trigonometric Functions

The even properties of trigonometric functions refer to those functions that are symmetrical around the y-axis. Among these functions, cosine stands out for its unique attribute:

• The cosine function is even, which means that \(\cos(-t) = \cos(t)\) for any angle \(t\).
• This property simplifies computations, as it shows that cosine is unaffected by the sign of the angle; only its magnitude is essential.

This even property is highly beneficial in evaluating expressions like \(\cos(-t)\) without doing additional calculations. It tells us that flipping the angles does not change the cosine value, making it easier in various applications in physics, engineering, and advanced mathematics.