half angle formula to find exact value
cos 112.5 degree
Why did the angle go to the party alone?
Because it couldn't find its other half!
But don't worry, I can help you find the exact value of cos 112.5 degrees using the half-angle formula!
The half-angle formula for cosine is given by:
cos(x/2) = ±√[(1 + cos(x))/2]
Let's apply this formula to find the exact value of cos(112.5 degrees):
Step 1: To use the half-angle formula, we need to find the value of cos(225 degrees) first.
Step 2: Using the half-angle formula, we have:
cos(112.5 degrees) = ±√[(1 + cos(225 degrees))/2]
Step 3: Now, let's find the value of cos(225 degrees):
cos(225 degrees) = - √2/2
Step 4: Plugging in the value of cos(225 degrees) into the half-angle formula:
cos(112.5 degrees) = ±√[(1 - √2/2)/2]
So, the exact value of cos(112.5 degrees) using the half-angle formula is ±√[(1 - √2/2)/2].
Please note that the ± sign indicates there are two possible solutions, since cos(x) has positive and negative values in different quadrants.
To find the exact value of cos 112.5 degrees using the half-angle formula, follow these steps:
Step 1: Convert 112.5 degrees into radians. Recall that π radians is equivalent to 180 degrees.
112.5 degrees * (π/180 degrees) = 1.9635 radians (rounded to 4 decimal places)
Step 2: Apply the half-angle formula for cosine:
cos(x/2) = ± √[(1 + cos(x)) / 2]
Step 3: Plug in the value of x = 2 * 1.9635 (since it's a half-angle):
cos(1.9635 / 2) = ± √[(1 + cos(1.9635)) / 2]
Step 4: Calculate the value of cos(1.9635):
Using a scientific calculator, cos(1.9635) = -0.8021 (rounded to 4 decimal places)
Step 5: Substitute the value of cos(1.9635) into the half-angle formula:
cos(1.9635 / 2) = ± √[(1 + (-0.8021)) / 2]
Step 6: Simplify the expression:
cos(1.9635 / 2) = ± √[(1 - 0.8021) / 2]
= ± √[0.1979 / 2]
= ± √0.09895
= ± 0.3147 (rounded to 4 decimal places)
Therefore, the exact value of cos 112.5 degrees using the half-angle formula is ± 0.3147.
To find the exact value of cos 112.5 degrees using the half-angle formula, we first need to determine the angle that is half of 112.5 degrees.
The half-angle formula for cosine is given as:
cos(theta/2) = ±√[(1 + cos(theta))/2]
In this case, we want to find cos(112.5/2) = cos(56.25) degrees.
To find the exact value, we can use the following steps:
1. Express 56.25 degrees in terms of radians:
- To convert degrees to radians, use the formula: radians = degrees * (pi/180)
- Plugging in the value, we have 56.25 degrees * (pi/180) = 0.981747704246812 rad
2. Substitute the angle in radians into the half-angle formula:
- cos(56.25) = ±√[(1 + cos(112.5))/2]
3. Calculate cos(112.5) using the basic cosine formula:
- cos(112.5) = cos(135 - 22.5) = -cos(22.5) (since cos(135) = 0)
- We can look up the exact value of cos(22.5) or calculate it using another angle-reduction formula.
Alternatively, we can convert 22.5 degrees to radians (22.5 * (pi/180)) = 0.39269908169872414 rad and find its cosine using a scientific calculator, which is approximately √(2 + √(2))/2 ≈ 0.92388.
4. Substitute the value of cos(22.5) into the half-angle formula:
- cos(112.5) ≈ -0.92388
5. Evaluate the half-angle formula using the value of cos(112.5):
- cos(56.25) = ±√[(1 + cos(112.5))/2] ≈ ±√[(1 - 0.92388)/2]
- Simplifying the expression, we get cos(56.25) ≈ ±√[(0.07612)/2] ≈ ±√(0.03806)
- Taking the positive square root, we have cos(56.25) ≈ √(0.03806)
Therefore, the exact value of cos 112.5 degrees using the half-angle formula is approximately √(0.03806).
recall cos 2x = 2cos^2 x - 1
2cos^2 x = cos 2x + 1
cos^2 x = (cos 2x + 1)/2
cos^2 (112.5°) = (cos 225° + 1)/2
now cos 225 = cos(180 + 45)
= cos180cos45 - sin180sin45
= -1(√2/2) - 0
= -√2/2
cos^2 112.5 = (-√2/2 + 1)/2= (2-√2)/4
cos(112.5) = ± √(2-√2)/2
but 112.5 is in quadrant II
so
cos(112.5°) = -(√(2-√2) )/2