Problem 77 - Algebra 2

Question

Which of these angles have a sine of about \(-0.6 ?\) $$\begin{array}{llll} \text { I. } {1.143 .1^{\circ}} & {\text { II. } 216.9^{\circ}} & {\text { III. } 323.1^{\circ}}\end{array}$$ A. I and II only B. I and III only C. II and III only D. I, II, and III

Step-by-Step Solution

Answer

The angles that have sine values of about -0.6 are angles II and III. So the answer is C. II and III only.

Step 1: Calculate the Sine of Angle I

Calculate the sine of the first angle, 143.1 degrees. To do this, input the value into the sine function on your calculator. Make sure your calculator is in degree mode. The result is approximately 0.61, which is not close to -0.6.

Step 2: Calculate the Sine of Angle II

Calculate the sine of the second angle, 216.9 degrees and compare it to the required value. The sine of this angle is approximately -0.58, which is close to -0.6.

Step 3: Calculate the Sine of Angle III

Calculate the sine of the third angle, 323.1 degrees. The sine of this angle is approximately -0.59, which is also close to -0.6.

Key Concepts

Sine Function

The sine function is a cornerstone concept in trigonometry, expressing the relationship between the length of the opposite side to an angle and the hypotenuse in a right-angled triangle. Generally denoted by \( \sin(\theta) \), where \( \theta \) signifies the angle, it offers important insights not just in triangles but in wave patterns, oscillations, and rotations. It's crucial to know that the sine function can take on values ranging from -1 to 1.
To understand values like -0.6, consider it in the context of the Unit Circle—a circle with a radius of 1 centered at the origin of a graph. The coordinate at any angle \( \theta \) on this circle can represent this angle's sine value. Angles with a sine of about -0.6 are in the third and fourth quadrants, where the sine takes negative values. This understanding aids in addressing questions about angles yielding specific sine values.

Angle Calculation

Angle calculation in trigonometry involves determining either the exact angle or function value. In practice, having the angle, like 143.1 degrees, lets us compute its sine value using calculators set to degree mode. This process involves:

• Inputting the specific angle
• Selecting the sine function (usually denoted as \( \sin \))
• Reading the output value

For instance, at 216.9 degrees, the sine is estimated to be -0.58, showing its proximity to -0.6. Thus, understanding how to use these functions is essential in identifying or confirming correct angles for particular sine values. Remember, the role of a calculator is to rapidly compute values that might otherwise involve more complex mathematical tables or estimations.

Degree Mode

Using Degree Mode in a calculator is vital for correctly solving problems involving angles expressed in degrees. Degrees, one of the most common angle measurement units, divide a circle into 360 equal parts. Calculators can be set to different modes such as Degrees or Radians, which is another unit of measurement. For our exercises:
Ensure your calculator is in Degree Mode, indicated often by a 'D' sign on the screen. To check or change it:

• Look for a mode button
• Toggle between Deg (degree) and Rad (radian)
• Select Deg for work involving angles measured in degrees

Ensuring correct mode prevents errors in results, as different modes yield different function values for the same input angle. Thus, mastering this setting is critical when performing angle-related calculations in degrees effectively.