Calculate The Exact Value of Cos 225 Degrees
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Calculating the exact value of cos 225 degrees involves understanding trigonometric identities and reference angles. This guide explains the process step-by-step, including how to use the cosine function for angles beyond 180 degrees.
How to Calculate cos 225 Degrees
The cosine of an angle in the third quadrant (180° to 270°) is negative because both the x-coordinate and y-coordinate of the point on the unit circle are negative. To find cos 225°, we can use the reference angle and the properties of the cosine function.
Key Steps:
- Identify the quadrant of the angle (225° is in the third quadrant).
- Find the reference angle by subtracting 180° from the given angle.
- Recall that cosine is negative in the third quadrant.
- Calculate the cosine of the reference angle.
- Apply the sign based on the quadrant.
For 225°, the reference angle is 225° - 180° = 45°. The cosine of 45° is √2/2. Since we're in the third quadrant where cosine is negative, cos 225° = -√2/2.
Formula Used
The exact value of cos θ can be calculated using trigonometric identities. For angles in the third quadrant:
cos θ = -cos(θ - 180°)
Where θ is the angle in degrees, and θ - 180° is the reference angle.
For θ = 225°:
cos 225° = -cos(45°)
cos 45° = √2/2, so cos 225° = -√2/2 ≈ -0.7071.
Worked Example
Let's calculate cos 225° step-by-step:
- Identify the quadrant: 225° is in the third quadrant (180° to 270°).
- Find the reference angle: 225° - 180° = 45°.
- Recall that cosine is negative in the third quadrant.
- Calculate cos 45°: √2/2 ≈ 0.7071.
- Apply the sign: cos 225° = -√2/2 ≈ -0.7071.
Final Result:
cos 225° = -√2/2 ≈ -0.7071
Interpreting the Result
The negative value of cos 225° indicates that the x-coordinate of the point on the unit circle is negative. This makes sense because 225° is in the third quadrant where both x and y coordinates are negative.
The exact value -√2/2 is a precise mathematical representation, while the approximate value -0.7071 is useful for practical calculations. Both forms are correct depending on the context.
Frequently Asked Questions
Why is cos 225° negative?
cos 225° is negative because 225° is in the third quadrant (180° to 270°), where both the x-coordinate and y-coordinate of the point on the unit circle are negative. Cosine represents the x-coordinate, so it's negative in this quadrant.
How do I calculate the exact value of cos 225°?
To calculate cos 225° exactly, find the reference angle (225° - 180° = 45°), then take the negative of the cosine of the reference angle: cos 225° = -cos 45° = -√2/2.
What is the approximate value of cos 225°?
The approximate value of cos 225° is -0.7071. This is derived from the exact value -√2/2 ≈ -0.7071.
Can I use a calculator to find cos 225°?
Yes, you can use a scientific calculator to find cos 225°. Just enter 225 and press the cosine button. The calculator will return the same result as the manual calculation: -√2/2 ≈ -0.7071.
What is the reference angle for 225°?
The reference angle for 225° is 45°. It's found by subtracting 180° from 225° (225° - 180° = 45°).