If cos25+sin25=k then the value os cos20 is ?
Cos25°+sin25°=p cos50°=?
To find the value of cos20, we can use the identity cos(A-B) = cosA * cosB + sinA * sinB.
Let's express cos25 as a subtraction of angles: cos25 = cos(45-20).
Using the identity, we have:
cos(45-20) = cos45 * cos20 + sin45 * sin20.
Since cos45 = sin45 = √2/2 and sin20 can be expressed in terms of cos20, we can substitute these values into the equation:
cos(45-20) = (√2/2) * cos20 + (√2/2) * sin20.
But we know that sin20 = 1 - cos^2(20), so we can substitute sin20:
cos(45-20) = (√2/2) * cos20 + (√2/2) * (1 - cos^2(20)).
Since cos^2(20) = 1 - sin^2(20), we can substitute another value:
cos(45-20) = (√2/2) * cos20 + (√2/2) * (1 - (1 - cos^2(20))).
Now let's simplify the equation:
cos(45-20) = (√2/2) * cos20 + (√2/2) * (cos^2(20)).
We also know that cos^2(20) = 1 - sin^2(20) = 1 - (√2/2) * cos20. Substituting this value into the equation:
cos(45-20) = (√2/2) * cos20 + (√2/2) * (1 - (√2/2) * cos20).
Now let's further simplify the equation:
cos(45-20) = (√2/2) * cos20 + (√2/2) - (√2/4) * cos20.
Combining like terms:
cos(45-20) = (3√2/4) * cos20 + (√2/2).
We were given that cos25 + sin25 = k. Since cos(45-20) = cos25 + sin25, we can equate the two equations:
(3√2/4) * cos20 + (√2/2) = k.
Solving for cos20:
(3√2/4) * cos20 = k - (√2/2).
cos20 = (4/3√2) * (k - (√2/2)).
Therefore, the value of cos20 is given by cos20 = (4/3√2) * (k - (√2/2)).
To solve for the value of cos20, we can use the given equation cos25 + sin25 = k.
First, let's manipulate the equation to isolate the cos25 term:
cos25 = k - sin25
Next, we will use the identity cos(a - b) = cos(a)cos(b) + sin(a)sin(b):
cos25 = k - sin25
cos(25 - 25) = k - sin25
cos^2(25) - sin^2(25) = k - sin25
cos^2(25) - (1 - cos^2(25)) = k - sin25
2cos^2(25) - 1 = k - sin25
Since cos20 is related to cos25, we can make use of another identity, namely cos(a - b) = cos(a)cos(b) + sin(a)sin(b):
cos25 = cos(20 + 5)
cos(20)cos(5) + sin(20)sin(5) = k - sin25
cos^2(20) + sin^2(20) + 2sin(20)sin(5) = k - sin25
1 + 2sin(20)sin(5) = k - sin25
Now, we can substitute cos^2(25) = 1 - sin^2(25) into the equation:
2cos^2(25) - 1 = k - sin25
2(1 - sin^2(25)) - 1 = k - sin25
2 - 2sin^2(25) - 1 = k - sin25
1 - 2sin^2(25) = k - sin25
2sin^2(25) - 1 = sin25 - k
By comparing this equation with the expression derived earlier, we can see that:
2sin(20)sin(5) = sin25 - k
Now, we can solve for sin25:
sin25 = 2sin(20)sin(5) + k
Since the value of k is not given, we cannot directly determine the value of sin25. Therefore, we cannot find the precise value of cos20 using the given information.
cos20 = cos(70-50) = cos70cos50 + sin70sin50
cos25+sin25 = √2 sin70 = k, so
sin70 = k/√2
cos70 = √(2-k^2) / √2
cos^2 25 + 2sin25cos25 + sin^2 25 = k^2
1 + sin50 = k^2
sin50 = k^2-1
cos50 = k√(2-k^2)
so, now we have
cos 20 = k√(2-k^2)√(2-k^2) / √2 + k(k^2-1)/√2
= (k(2-k^2) + k(k^2-1))/√2
= k/√2