Problem 163 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \sin 310^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 310^{\circ} \approx -0.766 \).

Step 1: Convert the Angle into Radians

To evaluate \( \sin(310^{\circ}) \) using a calculator, the angle must first be converted from degrees to radians. Use the conversion formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). Hence, \( 310^{\circ} \times \frac{\pi}{180} = \frac{31\pi}{18} \) radians.

Step 2: Recognize Angle Significance

310° is in the fourth quadrant of the unit circle. Here, the sine function is negative, implying that \( \sin 310^{\circ} \) is negative.

Step 3: Use a Calculator to Find the Sine Value

Enter \( 310^{\circ} \) into the graphing calculator making sure it's set to degree mode, and find \( \sin 310^{\circ} \). The value obtained will be approximately -0.766. Ensure that you have used a negative angle sign in case of the fourth quadrant.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It is one of the primary trigonometric functions, alongside cosine and tangent. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. This function helps describe the oscillatory nature of waves.

• For the unit circle, which has a radius of 1, sine represents the y-coordinate of a point on the circle.
• The sine function varies between -1 and 1 as angles change from 0 to 360 degrees (or 0 to 2π radians).

An important aspect of the sine function is its periodicity. Its values repeat in cycles, meaning after every 360 degrees or 2π radians, the sine values return to their initial state. This property is crucial when analyzing periodic phenomena such as sound waves or tides.

Angle Conversion

Angle conversion is essential when working with trigonometric functions like sine. Angles are commonly measured in degrees and radians. Converting between these units is often necessary, especially in calculations using mathematical software or calculators.

• The conversion formula is: radians = degrees × \( \frac{\pi}{180} \).
• This formula allows you to convert any angle from degrees to radians.

For instance, to convert 310 degrees to radians, you calculate: 310° × \( \frac{\pi}{180} \) = \( \frac{31\pi}{18} \).Understanding angle conversion is vital as many mathematical applications, like calculus and physics, predominantly use radians. Calculators often require input in radians for correct trigonometric calculations.

Unit Circle

The unit circle is a powerful tool in trigonometry. It provides a geometric representation of the relationships between angles and trigonometric functions. The unit circle has a radius of 1 and is centered at the origin of the coordinate plane.

• Each point on the unit circle corresponds to a specific angle measured from the positive x-axis.
• The coordinates of these points are \( (\cos(\theta), \sin(\theta)) \), where \( \theta \) is the angle.

Knowing which quadrant an angle is in can determine the sign of its sine value. In the case of 310 degrees, it lies in the fourth quadrant. Here, the values of sine are negative. This knowledge helps quickly indicate whether the sine is positive or negative without actual computation. The unit circle thus offers a visual method to understand and memorize trigonometric concepts intuitively.

Graphing Calculator

A graphing calculator is an advanced tool that assists in performing complex mathematical calculations, including those involving trigonometric functions. When evaluating functions like sine, it's important to ensure the calculator settings are appropriate.

• Ensure the calculator is set to degree mode when working with angles in degrees, or switch to radian mode for radian measures.
• Input the angle correctly to find its sine value directly.

For example, entering 310° once your calculator is in degree mode allows for direct computation of \( \sin 310^{\circ} \). The result should be approximately -0.766 when the sine function is applied correctly. Graphing calculators not only compute values but can also visualize functions and explore their behavior, aiding deeper understanding and analysis.