Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \sin 280^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 280^{\circ} \approx -0.9848 \).

Step 1: Identify the Angle

First, identify the angle we are working with, which is given as \( 280^{\circ} \). We need to find the sine of this angle.

Step 2: Reference Angle Calculation

Calculate the reference angle. The reference angle for any angle \( \theta \) in standard position is the acute angle that the terminal side of \( \theta \) makes with the x-axis. Since \( 280^{\circ} \) is in the fourth quadrant, the reference angle is \( 360^{\circ} - 280^{\circ} = 80^{\circ} \).

Step 3: Determine Sine in the Quadrant

In the fourth quadrant, the sine of an angle is negative. Therefore, \( \sin 280^{\circ} = -\sin 80^{\circ} \).

Step 4: Use a Calculator

Using a scientific calculator, find \( \sin 80^{\circ} \). Enter \( 80 \, ^{\circ} \) and calculate the sine, which yields \( \sin 80^{\circ} \approx 0.9848 \). Thus, \( \sin 280^{\circ} = -0.9848 \).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often abbreviated as "sin." It relates an angle of a right triangle to the ratios of two of its sides. Specifically, for a given angle \( \theta \), the sine function computes the ratio of the opposite side over the hypotenuse. Mathematically, this is expressed as: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]. The sine function is cyclic, with a periodicity of \( 360^{\circ} \) or \( 2\pi \) radians. This means that repeating cycles occur every \( 360^{\circ} \). The function spans a range from -1 to 1, capturing the essence of how angles relate to side ratios in right triangles across the unit circle.It's crucial to remember that the sine function changes its sign based on the quadrant of the angle. In the case of the angle \( 280^{\circ} \), which lies in the fourth quadrant, the sine value is negative, demonstrating the cyclical nature alongside its positive and negative values alternating by quadrants.

Reference Angle

A reference angle is always a positive acute angle (less than \( 90^{\circ} \)) that helps simplify the trigonometric calculations and relate non-acute angles to their acute angle counterparts. To find the reference angle:

• If an angle is in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from \( 180^{\circ} \).
• In the third quadrant, subtract \( 180^{\circ} \) from the angle.
• In the fourth quadrant, subtract the angle from \( 360^{\circ} \).

For \( 280^{\circ} \), the angle is in the fourth quadrant, so its reference angle is calculated as \( 360^{\circ} - 280^{\circ} = 80^{\circ} \). This reference angle simplifies computing sine, cosine, or tangent, since you can use their acute-angle equivalents.

Quadrant Analysis

Understanding quadrants in trigonometry is essential to determine the positive or negative value of trigonometric functions. The coordinate plane is divided into four quadrants, each affecting the signs of the trigonometric functions:

• **First Quadrant**: All trigonometric functions (sin, cos, tan) are positive.
• **Second Quadrant**: Sine is positive; cosine and tangent are negative.
• **Third Quadrant**: Tangent is positive; sine and cosine are negative.
• **Fourth Quadrant**: Cosine is positive; sine and tangent are negative.

Analyzing which quadrant an angle lies in helps to assess the correct sign of the function value. For example, since \( 280^{\circ} \) is located in the fourth quadrant, \( \sin 280^{\circ} = -\sin 80^{\circ} \), confirming the sine function is negative due to residing in the fourth quadrant—highlighting how quadrant location influences the trigonometric sign and functionality.