Solve Cos 315 Without Calculator
'/%3E%3Ccircle cx='48' cy='39' r='19' fill='%23f2c9a5'/%3E%3Cpath d='M27 35c3-16 39-17 42 2-8-8-27-9-42-2z' fill='%23374151'/%3E%3Cpath d='M21 86c5-24 49-28 56 0z' fill='%232563eb'/%3E%3Ccircle cx='41' cy='41' r='2' fill='%23111827'/%3E%3Ccircle cx='55' cy='41' r='2' fill='%23111827'/%3E%3Cpath d='M42 52c4 3 9 3 13 0' stroke='%2392400e' stroke-width='3' fill='none' stroke-linecap='round'/%3E%3C/svg%3E)
Reviewed by Calculator Editorial Team
Calculating the cosine of 315 degrees without a calculator requires understanding of the unit circle and reference angles. This guide explains the method, provides a step-by-step calculation, and includes a worked example.
How to calculate cos 315° without a calculator
The cosine of 315 degrees can be found using the unit circle and reference angles. Here's the method:
- Identify the reference angle for 315°
- Determine the quadrant where 315° is located
- Use the cosine value of the reference angle
- Apply the sign based on the quadrant
Formula: cos(360° - θ) = cosθ
Since 315° is 45° less than 360°, cos(315°) = cos(45°)
The cosine of 315 degrees is equal to the cosine of 45 degrees because they are related by the cosine function's periodicity.
Step-by-step calculation
-
Identify the reference angle
315° is located in the fourth quadrant. The reference angle is calculated as:
Reference angle = 360° - 315° = 45°
-
Determine the quadrant
315° is in the fourth quadrant (270° to 360°). In this quadrant, cosine values are positive.
-
Use the cosine of the reference angle
The cosine of 45° is a well-known value:
cos(45°) = √2/2 ≈ 0.7071
-
Apply the sign based on the quadrant
Since 315° is in the fourth quadrant where cosine is positive:
cos(315°) = +cos(45°) = √2/2 ≈ 0.7071
Worked example
Let's calculate cos(315°):
- Reference angle = 360° - 315° = 45°
- 315° is in the fourth quadrant (cosine positive)
- cos(45°) = √2/2 ≈ 0.7071
- Therefore, cos(315°) = +√2/2 ≈ 0.7071
The exact value is √2/2, and the approximate decimal value is 0.7071.
FAQ
- Why is cos(315°) equal to cos(45°)?
- Because 315° is 45° less than 360°, and the cosine function has a period of 360°, making cos(315°) = cos(45°).
- Is the cosine of 315° positive or negative?
- Positive, because 315° is in the fourth quadrant where cosine values are positive.
- What is the reference angle for 315°?
- The reference angle is 45°, calculated as 360° - 315° = 45°.
- Can I use this method for other angles?
- Yes, this method works for any angle by finding its reference angle and applying the appropriate sign based on the quadrant.