Question: How to find the exact value of cosine of π/4?
The cosine of π/4 can be found by using the trigonometric identity and values of common angles. The exact value of cosine π/4 is √2/2.
We can start by using the cosine addition or subtraction formula, which states that:
cos(A ± B) = cos(A) * cos(B) ∓ sin(A) * sin(B)
In this case, we have A = B = π/4, so we can rewrite the formula as:
cos(π/4 ± π/4) = cos(π/4) * cos(π/4) ∓ sin(π/4) * sin(π/4)
Simplifying further, we get:
cos(π/2) = cos^2(π/4) – sin^2(π/4)
Since sin(π/4) and cos(π/4) are equal, we can substitute these values and calculate:
cos(π/2) = cos^2(π/4) – cos^2(π/4)
cos(π/2) = 2cos^2(π/4) – 1
Now we have an equation with only one variable, cos(π/4). Let’s solve for cos(π/4):
2cos^2(π/4) – 1 = 0
cos^2(π/4) = 1/2
Taking the square root of both sides, we find:
cos(π/4) = ±√(1/2)
Since cosine is positive in the first and fourth quadrants, we consider the positive value:
Therefore, the exact value of cosine of π/4 is √2/2.
Frequently Asked Questions:
Q1: What is the cosine of π/3?
The exact value of cosine π/3 is 1/2.
Q2: How do I calculate the cosine of π/6?
The cosine of π/6 can be found by using the exact value of cosine π/3. After halving the angle, the cosine π/6 also simplifies to 1/2.
Q3: What is the cosine of π/2?
The cosine of π/2 is equal to zero.
Q4: How can I find the value of cosine 2π/3?
By utilizing the symmetry of the unit circle, we can determine that the cosine of 2π/3 is -1/2.
Q5: What is the cosine of π?
The cosine of π is equal to -1.
Q6. How do I find the cosine of 3π/4?
Using the properties of quadrants and reference angles, we can determine that the cosine of 3π/4 is -√2/2.
Q7: What is the cosine of 5π/6?
The cosine of 5π/6 simplifies to -1/2.
Q8: Can cosine be greater than 1?
No, cosine can never be greater than 1. The cosine function produces values between -1 and 1 for any angle.
Q9: How to find the cosine of an obtuse angle?
To find the cosine of an obtuse angle, you can use the cosine of its supplementary angle. Subtract the obtuse angle from 180 degrees or π radians and calculate the cosine of the resulting acute angle.
Q10: What is the cosine of an angle in radians?
The cosine of an angle in radians represents the ratio of the length of the adjacent side to the length of the hypotenuse in a unit circle.
Q11: Can cosine be negative?
Yes, cosine can be negative. It is negative in the second and third quadrants of the unit circle.
Q12: How can I find the value of cosine for non-standard angles?
For non-standard angles, you can use a calculator or table of trigonometric values to find the approximate value of cosine.