how would you express cos(75 as a single fraction?
im assuming oyu saying cos75 degree as a simple fraction;
cos(75)=cos(45+30)
recall the formula cos(a+b)=cos a cosb-
sin a sin b
cos(45+30)=cos(30)sin(45)-
sin(30)sin(45)
cos(45+30)=((sqrt(3))/2)((sqrt(2))/2)-…
cos(45+30)
=(sqrt(6)- sqrt(2))/4
To express cos(75) as a single fraction, we can use the angle sum formula for cosine, which states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
Since 75 degrees can be expressed as the sum of two angles, such as 45 degrees and 30 degrees, we can rewrite cos(75) as cos(45 + 30).
Using the angle sum formula, we have:
cos(75) = cos(45 + 30)
= cos(45)cos(30) - sin(45)sin(30)
Now, we can use the values from the unit circle to simplify further:
cos(45) = √2 / 2
cos(30) = √3 / 2
sin(45) = √2 / 2
sin(30) = 1 / 2
Substituting these values, we get:
cos(75) = (√2 / 2)(√3 / 2) - (√2 / 2)(1 / 2)
= (√6 / 4) - (√2 / 4)
= (√6 - √2) / 4
Therefore, cos(75) can be expressed as the single fraction (√6 - √2) / 4.
To express cos(75) as a single fraction, we can utilize the trigonometric identity: cos(2θ) = 2cos²(θ) - 1.
First, let's rewrite 75 as a sum or difference of angles whose cosine values are known. Since 75 is not directly an angle with a known cosine value, we can express it as the sum of two angles: 75 = 45 + 30.
Therefore, cos(75) = cos(45 + 30).
Next, we use the identity for the cosine of the sum of two angles: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
Applying this identity, we have:
cos(75) = cos(45 + 30)
= cos(45)cos(30) - sin(45)sin(30).
Now, we can use the known values of cos(45) = 1/√2 and sin(45) = 1/√2, as well as cos(30) = √3/2 and sin(30) = 1/2.
Substituting these values into the expression, we get:
cos(75) = (1/√2) * (√3/2) - (1/√2) * (1/2).
Simplifying further, we have:
cos(75) = √3 / (2√2) - 1 / (2√2).
To rationalize the denominator, we multiply both the numerator and denominator by √2:
cos(75) = (√3 * √2) / (2 * √2 * √2) - (1 * √2) / (2 * √2 * √2).
Simplifying again, we obtain:
cos(75) = √6 / (2 * 2) - √2 / (2 * 2).
Finally, we simplify further to arrive at the single fraction form:
cos(75) = √6 / 4 - √2 / 4.
Therefore, cos(75) expressed as a single fraction is (√6 - √2) / 4.