Find the value of two sin 75
degrees cos 75 degrees without using a calculator.
To find the value of this
expression, we can use the formula two sin 𝐴 cos 𝐴 is equal to sin of two 𝐴. And since we have two sin 75
degrees cos 75 degrees, 𝐴 must be equal to 75 degrees. So we can plug in 75 for 𝐴. So now we can take two times 75 and
get the sin of 150 degrees.
So without using the calculator, we
can imagine 150 degrees is in the second quadrant. And now to find the value at the
sin of 150 degrees, let’s first look at the reference angle. So that’s the terminal side that’s
close to the 𝑥-axis. So here we’re at 180 degrees, so if
we’re at 150, this angle must be 30 degrees. And knowing our values with the
unit circle or using it from the special right triangle, the 30-60-90 triangle, we
can remember that the sin of 30 degrees is one-half and the cos of 30 degrees is
square root three over two.
Now cos of 𝜃 will represent our
𝑥-value and the sin of 𝜃 will represent our 𝑦-value. However, 𝑥 and 𝑦 aren’t both
positive in quadrant two. In quadrant two, 𝑦 is positive but
𝑥 is negative. So that means that the cos at 150
degrees would be negative square root three over two and the sin would be positive
one-half. So that means we can use the sin of
150 degrees to be the exact same as the sin of 30 degrees, one-half.