Problem 24 - Algebra 2

Question

Find the exact values of the cosine and sine of each angle. Then find the decimal values. Round your answers to the nearest hundredth. $$ -240^{\circ} $$

Step-by-Step Solution

Answer

The cosine of -240 degrees is -0.5 and the sine is approximately 0.87

Step 1: Convert the Angle to Positive

Since -240 degrees is a negative angle, to simplify the problem, it can be converted into a positive angle by adding 360 degrees to -240 degrees. Hence, the equivalent positive angle is \(360^{\circ} - 240^{\circ} = 120^{\circ}\).

Step 2: Locate the Angle on the Unit Circle

Now, locate the 120 degrees on the unit circle. 120 degrees is in the second quadrant.

Step 3: Determine the Cosine and Sine Values

In the second quadrant, the reference angle is \(180^{\circ} - 120^{\circ} = 60^{\circ}\). Using the 60 degrees reference angle, the corresponding cosine and sine values can be retrieved from the unit circle. Cosine is negative and sine is positive in the second quadrant. Thus, \(\cos(120^{\circ}) = -\cos(60^{\circ}) = -0.5\) and \(\sin(120^{\circ}) = \sin(60^{\circ}) = \sqrt{3}/2\).

Step 4: Convert to Decimal Values

Finally, we convert the exact values obtained in Step 3 into decimal up to the nearest hundredth. Therefore, \(\cos(-240^{\circ}) = -0.5\) and \(\sin(-240^{\circ}) = \sqrt{3}/2 \approx 0.87\).

Key Concepts

Cosine and Sine

Cosine (\(\cos\) ) and sine (\(\sin\) ) are fundamental trigonometric functions used to relate angles to ratios of sides of a right-angled triangle. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse, while the sine is the ratio of the length of the opposite side to the hypotenuse. The relationships can be remembered as:

• Cosine: \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• Sine: \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)

These functions are also integral to understanding rotations and movements in geometric space. They range between -1 and 1 for any angle and have periodic properties, repeating every 360 degrees or \(2\pi\) radians. For angles found on the unit circle, we look at how these ratios translate into the Cartesian coordinate plane, marking out specific points that help us determine their exact values.

Unit Circle

The unit circle is a conceptual tool used in trigonometry, defined as a circle with a radius of 1, centered at the origin \((0, 0)\) in the coordinate plane. Each point on the unit circle corresponds to the cosine and sine of a specific angle, forming coordinates \((\cos \theta, \sin \theta)\). This allows for easy visualization of trigonometric functions and angles.

• The unit circle helps visualize how cosine and sine values repeat in cycles as the angle increases.
• It designates key angles, like 0°, 90°, 180°, and 270°, and calculates their cosine and sine values.
• Each quadrant of the circle alters the sign of these functions, e.g., cosine is negative in the second and third quadrants.

When finding \(\cos\) and \(\sin\) for an angle like 120° or \(-240°\), the unit circle is a guiding tool as it visually shows the positions and helps derive the values of trigonometric functions accurately.

Reference Angles

Reference angles are the angles formed with the x-axis in each quadrant, providing a simple way to calculate trigonometric functions for any angle. For an angle in standard position, its reference angle is the acute angle it makes with the x-axis. This angle is always less than 90 degrees.

• Reference angles allow for the simplification of calculations by considering symmetry in the circle.
• They help to determine positive or negative signs for cosine and sine values depending on the quadrant.
• For example, -240° converts to 120°, its reference angle becomes 60° (\(180° - 120°\)), which is a well-known angle.

Using reference angles, we can use known values like those for 30°, 45°, or 60° to find the cosine and sine of less common or larger angles efficiently. It's an effective method to break down problems into simpler, easier-to-calculate components in trigonometry.