Find the value of sin negative 60 degrees multiplied by cos 30 degrees plus the tan of 57 degrees over the cot of 33 degrees, giving the answer in its simplest form.
We will answer this question by looking at the two parts of our expression separately. We begin by recalling that since the sine function is an odd function, the sin of negative 𝜃 is equal to negative sin 𝜃. This means that the sin of negative 60 degrees is equal to negative sin of 60 degrees. The first part of our expression can be rewritten as negative sin of 60 degrees multiplied by cos of 30 degrees.
We know that 30 degrees and 60 degrees are two of our special angles. The sin of 60 degrees and cos of 30 degrees are both equal to root three over two. This means that this part of our expression is equal to negative root three over two multiplied by root three over two. Since root three multiplied by root three is three and two multiplied by two is four, this simplifies to negative three-quarters.
Let’s now consider the second part of our expression. 57 degrees and 33 degrees are complementary angles as they sum to 90 degrees. This means that 57 degrees is equal to 90 degrees minus 33 degrees. This means that we can rewrite the numerator as the tan of 90 degrees minus 33 degrees. Next, we recall one of the cofunction identities. The tan of 90 degrees minus 𝜃 is equal to the cot of 𝜃. This means that the tan of 90 degrees minus 33 degrees is equal to the cot of 33 degrees. We have the cot of 33 degrees divided by the cot of 33 degrees. This is equal to one.
The expression sin of negative 60 degrees multiplied by cos 30 degrees plus tan 57 degrees divided by cot 33 degrees is equal to negative three-quarters plus one, which is equal to one-quarter. The value of the expression in its simplest form is one-quarter.