Problem 24 - Algebra 2 and Trigonometry

Question

In \(18-27\) , express each given function value in terms of a function value of a positive acute angle (the reference angle). \(\cos 245^{\circ}\)

Step-by-Step Solution

Answer

\(\cos 245^{\circ} = -\cos 65^{\circ}\).

Step 1: Identify the Original Angle's Quadrant

The angle given is \(245^{\circ}\). First, determine which quadrant this angle is located in. Angles from \(180^{\circ}\) to \(270^{\circ}\) are in the third quadrant.

Step 2: Calculate the Reference Angle

The reference angle is found by subtracting the given angle from \(180^{\circ}\) because \(245^{\circ}\) is in the third quadrant. Calculate it as follows: \(245^{\circ} - 180^{\circ} = 65^{\circ}\).

Step 3: Determine the Cosine Sign in the Third Quadrant

In the third quadrant, the cosine of an angle is negative.

Step 4: Express in Terms of Reference Angle

Use the reference angle found in Step 2 to express \(\cos 245^{\circ}\) in terms of \(\cos 65^{\circ}\), considering the sign: \(\cos 245^{\circ} = -\cos 65^{\circ}\).

Key Concepts

Understanding Reference Angles

In trigonometry, a reference angle helps in simplifying the process of finding the trigonometric function values for any angle. The reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. Normally, this angle is always a positive acute angle, ranging from 0 to 90 degrees.
In practice, the reference angle assists in relating any angle to an easily understandable acute angle. To calculate the reference angle:

• For angles in the first quadrant (0° to 90°), the reference angle is the angle itself.
• In the second quadrant (90° to 180°), subtract the angle from 180°.
• In the third quadrant (180° to 270°), subtract 180° from the angle.
• For angles in the fourth quadrant (270° to 360°), subtract the angle from 360°.

This approach simplifies many trigonometric problems by reducing them to their acute counterparts.

The Cosine Function Simplified

The cosine function is one of the primary functions in trigonometry. It relates to the x-coordinate of a point on the unit circle for a given angle. Thus, \\[\cos\theta = \frac{x}{r}\]Where \(x\) is the horizontal coordinate and \(r\) is the radius, typically 1 in a unit circle.
Cosine is a periodic function, meaning it repeats its values in a regular pattern every 360 degrees. Its graph is a wave oscillating between -1 and 1.
Depending on the location of an angle within the quadrants, the cosine value can be either positive or negative. Specifically:

• Cosine is positive for angles in the first and fourth quadrants.
• It is negative in the second and third quadrants.

This behavior affects how we determine trigonometric function values, particularly when utilizing reference angles as a tool for simplification.

Navigating Quadrants in Trigonometry

Understanding quadrants in trigonometry is crucial as they dictate the sign of trigonometric functions. The coordinate plane is divided into four quadrants:

• The first quadrant covers angles from 0° to 90° and all trigonometric functions are positive.
• In the second quadrant (90° to 180°), sine is positive, while cosine and tangent are negative.
• The third quadrant (180° to 270°) features positive tangent and negative sine and cosine.
• Finally, the fourth quadrant (270° to 360°) has positive cosine and negative sine and tangent.

When evaluating function values of a specific angle, identifying the quadrant helps determine the sign of the trigonometric functions involved. This becomes particularly beneficial when expressing angles in terms of their reference angles, as seen with \\[\cos 245^{\circ} = -\cos 65^{\circ}\]Thus, knowing the quadrant provides a quick and reliable way to establish whether your derived trigonometric value should be positive or negative.