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