1)Find the exact value of sin 165 degrees.
Answer: (sqrt6 + sqrt2)/4
2) Simplify: (4m/5n^2)- (n/2m)
Answer: (8m^2-5n^3)/10n^2m
Thanks
The answer they gave you for the first one is not correct.
Here is the right way:
sin 165
= sin(120+45)
= sin120cos45 + cos120sin45
= sin60sin45 + (-cos60sin45)
= (√3/2)(√2/2) - (1/2)(√2/2)
= (√6 - √2)/4
(my answer checked with calculator)
To find the exact value of sin 165 degrees, we can use the sum-to-product formula for sine.
The sum-to-product formula states that sin(A + B) = sin A cos B + cos A sin B.
We can rewrite 165 degrees as the sum of two angles: 165 degrees = 90 degrees + 75 degrees.
Now, let's find the exact values of sin 90 degrees and sin 75 degrees.
We know that sin 90 degrees = 1, as it is the value for the sine of a right angle.
To find sin 75 degrees, we can use the trigonometric identity sin (A + B) = sin A cos B + cos A sin B.
Using 75 degrees = 45 degrees + 30 degrees, we can rewrite sin 75 degrees as sin (45 degrees + 30 degrees).
Since we know the exact values of sin 45 degrees and sin 30 degrees, we can substitute these values into the formula.
sin (45 degrees + 30 degrees) = sin 45 degrees cos 30 degrees + cos 45 degrees sin 30 degrees.
sin 45 degrees = sqrt(2)/2, and sin 30 degrees = 1/2. Similarly, cos 45 degrees = sqrt(2)/2, and cos 30 degrees = sqrt(3)/2.
Substituting these values into the formula, we get:
sin 75 degrees = (sqrt(2)/2) * (sqrt(3)/2) + (sqrt(2)/2) * (1/2).
Simplifying this expression, we get:
sin 75 degrees = sqrt(6)/4 + sqrt(2)/4.
So, the exact value of sin 165 degrees is:
sin 165 degrees = sin (90 degrees + 75 degrees) = sin 90 degrees cos 75 degrees + cos 90 degrees sin 75 degrees = 1 * (sqrt(6)/4 + sqrt(2)/4) = (sqrt(6) + sqrt(2))/4.
Therefore, the exact value of sin 165 degrees is (sqrt(6) + sqrt(2))/4.
Moving on to the second question:
To simplify (4m/5n^2) - (n/2m), we can start by finding a common denominator for the fractions.
The common denominator is 10m^2n^2.
We can rewrite (4m/5n^2) and (n/2m) with the common denominator:
(4m/5n^2) = (4m * 2m)/(5n^2 * 2m) = 8m^2/(10m^2n^2) = 8m^2/(10m^2n^2)
(n/2m) = (n * 5n^2)/(2m * 5n^2) = 5n^3/(10m^2n^2)
Now that we have a common denominator, we can subtract the two fractions:
(4m/5n^2) - (n/2m) = (8m^2/(10m^2n^2)) - (5n^3/(10m^2n^2))
To subtract these fractions, we must have the same denominator. In this case, the denominator is already the same, so we can subtract the numerators:
(8m^2 - 5n^3)/(10m^2n^2)
Thus, the simplified expression for (4m/5n^2) - (n/2m) is (8m^2 - 5n^3)/(10m^2n^2).
Hope this helps!