Find the exact value of three cos 75 minus three cos 15.
We could calculate the value of this expression by simply typing it into our calculator. However, as we are asked to find the exact value, we will use our knowledge of the special angles and their trigonometric values. We recall that the special angles are considered to be zero, 30, 45, 60, and 90 degrees. One way of recalling the sine and cosine of these angles is as follows. We write the numbers zero, one, two, three, and four in the sine row and write them in reverse order in the cosine row.
Next, we divide each of the numbers by four and, where possible, simplify. Zero divided by four is zero, two divided by four simplifies to one-half, and four divided by four is equal to one. We then take the square root of each of our values. The square root of zero is zero. Therefore, the sin of zero degrees and cos of 90 degrees are equal to zero. The square root of one is one. Therefore, the sin of 90 degrees and cos of zero degrees are equal to one. When square rooting a fraction, we can square root the numerator and denominator separately. This means that the square root of one-quarter is the same as the square root of one over the square root of four. This is equal to one-half. So the sin of 30 degrees and cos of 60 degrees is equal to one-half. The sin of 45 degrees and the cos of 45 degrees are both equal to one over root two. By rationalizing the denominator, we can also write this as root two over two. Finally, the sin of 60 degrees and cos of 30 degrees are equal to root three over two.
Let’s now consider how this helps us solve the problem in this question. We have the cos of 75 degrees and the cos of 15 degrees. We notice that 75 is equal to 45 plus 30 and 15 is equal to 45 minus 30. This means that we can rewrite the expression as three multiplied by cos of 45 plus 30 minus three multiplied by cos of 45 minus 30. We can then take out a factor of three as shown.
Next, we recall the trigonometric angle sum and difference identities. We know that the cos of 𝛼 plus 𝛽 is equal to cos 𝛼 cos 𝛽 minus sin 𝛼 sin 𝛽. This means that cos 45 plus 30 is equal to cos 45 multiplied by cos 30 minus sin 45 multiplied by sin 30. We also know that the cos of 𝛼 minus 𝛽 is equal to cos 𝛼 cos 𝛽 plus sin 𝛼 sin 𝛽. The cos of 45 minus 30 is therefore equal to cos 45 multiplied by cos 30 plus sin 45 multiplied by sin 30.
If we call these equations one and two, we can substitute the right-hand sides into our expression. When we subtract equation two from equation one, cos 45 cos 30 will cancel. Negative sin 45 sin 30 minus sin 45 sin 30 gives us negative two sin 45 sin 30. Our expression therefore simplifies to negative six sin 45 sin 30. We see from our table that the sin of 45 degrees is root two over two and the sin of 30 degrees is one-half. Substituting these into our expression, we have negative six multiplied by root two over two multiplied by one-half. We can divide the numerator and denominator by two. This leaves us with negative three root two over two. This is the exact value of three cos 75 degrees minus three cos 15 degrees.