Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \cos 485^{\circ} $$

Step-by-Step Solution

Answer

\( \cos 485^{\circ} \approx -0.5736 \)

Step 1: Find the Reference Angle

To find the reference angle for any angle, subtract or add multiples of 360° until the angle is between 0° and 360°. In this case, subtract 360° from 485°: \[ 485^{\circ} - 360^{\circ} = 125^{\circ}. \] Thus, the reference angle is 125°.

Step 2: Use the Reference Angle to Find the Cosine

Cosine is negative in the second quadrant (between 90° and 180°). The cosine of the reference angle is the same as the cosine of the angle itself, within its quadrant. Hence, \( \cos 485^{\circ} = \cos 125^{\circ} \).

Step 3: Calculate the Cosine

Use a calculator to find \( \cos 125^{\circ} \). Make sure your calculator is in degree mode. \[ \cos 125^{\circ} \approx -0.5736. \]

Step 4: Conclude the Result

Since \( \cos 125^{\circ} \approx -0.5736 \), it follows that \( \cos 485^{\circ} = -0.5736 \) to four decimal places.

Key Concepts

Understanding Reference Angles

A reference angle is a useful tool in trigonometry. It's the smallest angle that the terminal side of a given angle makes with the x-axis. To find it, reduce the angle to a value between 0° and 360° by adding or subtracting multiples of 360°. This sometimes means you're dealing with angles larger than a full circle, like in our example where we turned 485° into 125°. This process is really handy because it helps in understanding how angles larger than 360° relate to the standard 360° we commonly use.

Here's a step-by-step reminder on how to compute a reference angle:

• Identify if the angle is greater than 360° or a negative angle.
• If greater, subtract 360° until the angle is within 0° to 360°.
• If negative, add 360° instead until it falls in the desired range.

By doing this, you ensure that you can work with an angle which is familiar to you within the unit circle, and then apply it to calculate trigonometric function values like sine, cosine, and tangent.

The Cosine Function Explained

The cosine function, \( \cos \left( \theta \right) \), relates an angle to the x-coordinate of a point on the unit circle. Cosine values range from \(-1\) to \(1\). The function tells us how "far to the right or left" a point is on the circle.

Here's how it works in this specific example:

• First, determine the reference angle (here it was turned into 125°).
• Use the information about the quadrant to adjust the sign of your cosine value. Since 125° is in the second quadrant, where cosine values are negative, \, \(\cos 125^{\circ} = -0.5736\).
• This is why applying quadrant logic is crucial to finding whether cosine is positive or negative.

Remember:

• If the angle is in the first quadrant, cosine is positive.
• Second quadrant, cosine is negative.
• Third quadrant, cosine remains negative.
• Fourth quadrant, cosine is again positive.

This quadrant behavior is consistent due to the symmetrical nature of the circle and becomes very intuitive with practice.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. These quadrants help us understand the sign (positive or negative) of trigonometric functions like sine, cosine, and tangent for angles that fall within each part. As we move counterclockwise from the positive x-axis, each 90° section forms a quadrant. This leads to the following setup:

• First Quadrant (0° to 90°): All trigonometric functions are positive.
• Second Quadrant (90° to 180°): Sine is positive, cosine and tangent are negative.
• Third Quadrant (180° to 270°): Tangent is positive, sine and cosine are negative.
• Fourth Quadrant (270° to 360°): Cosine is positive, sine and tangent are negative.

In our exercise problem asking about the cosine of 485°, after finding the reference angle of 125°, we acknowledge this angle resides in the second quadrant where cosine is negative, confirming \(\cos 485^{\circ} = -0.5736\).

By understanding and memorizing the characteristics of these quadrants, you can accurately predict the sign of trigonometric functions for various angles.