How To Find Cos Of Obtuse Angle

How to Find the Cosine of an Obtuse Angle

Finding the cosine of an obtuse angle might seem tricky at first, but it's a straightforward process once you understand the unit circle and the properties of trigonometric functions. This article will guide you through different methods, ensuring you master this concept. Understanding cosine's relationship to the x-coordinate on the unit circle is key to solving this problem.

What is an Obtuse Angle?

Before diving into the calculations, let's define our terms. An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees. This means it falls in the second quadrant of the coordinate plane.

Methods for Finding the Cosine of an Obtuse Angle

We'll explore three main approaches: using the unit circle, applying the cosine function to the reference angle, and leveraging the calculator.

1. Using the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. Any point on the unit circle can be represented by the coordinates (cos θ, sin θ), where θ is the angle formed by the positive x-axis and the line connecting the origin to that point.

For obtuse angles (between 90° and 180°), the x-coordinate will be negative. Therefore, the cosine of an obtuse angle will always be negative. To find the exact value, you need to determine the x-coordinate of the point on the unit circle corresponding to your obtuse angle.

• Example: Find cos(120°). 120° lies in the second quadrant. The reference angle (the acute angle formed between the terminal side of the angle and the x-axis) is 180° - 120° = 60°. The x-coordinate at 60° is 1/2, but since we're in the second quadrant, the x-coordinate is -1/2. Therefore, cos(120°) = -1/2.

2. Using the Reference Angle

The reference angle simplifies the process. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. The cosine of the obtuse angle will have the same magnitude as the cosine of its reference angle, but with a negative sign.

• Steps:

  1. Find the reference angle: Subtract the obtuse angle from 180°.
  2. Find the cosine of the reference angle: Use your knowledge of common angles (30°, 45°, 60°) or a calculator.
  3. Add a negative sign: Because the obtuse angle lies in the second quadrant, where cosine is negative, add a negative sign to the cosine of the reference angle.

• Example: Find cos(150°).

  1. Reference angle: 180° - 150° = 30°
  2. cos(30°) = √3/2
  3. cos(150°) = -√3/2

3. Using a Calculator

Most scientific calculators can directly compute the cosine of any angle, including obtuse angles. Make sure your calculator is in degree mode. Simply input the angle and press the "cos" button. The calculator will automatically provide the correct (negative) value.

Important Considerations:

• Radians vs. Degrees: Ensure your calculator is set to the correct angle mode (degrees or radians) depending on the problem's requirements.
• Exact vs. Approximate Values: For common angles like 120°, 135°, or 150°, strive for exact values (using radicals). For other angles, an approximate decimal value from a calculator is acceptable.
• Understanding the Sign: Remember that the cosine of an obtuse angle is always negative. This stems from its position in the second quadrant of the Cartesian coordinate system.

By understanding these methods and practicing, you'll confidently calculate the cosine of any obtuse angle. Remember to utilize the unit circle for a visual understanding, the reference angle for simplified calculations, and a calculator for efficiency.