Find the value of cos 60 degrees times sin of 30 degrees minus sin of 60 degrees times tan of 60 degrees plus tan squared of 30 degrees without using a calculator.
Consider a right triangle that has one 30-degree angle and one 60-degree angle. We sometimes call it a 30-60-90 triangle. And in all 30-60-90 triangles, the side lengths are in ratio one, square root of 3, two. Two is the longest side. It corresponds to the hypotenuse, it’s right across from the right angle. The square root of three is the second longest side and is across from the 60-degree angle. And across from the 30-degree angle is the one. This information can help us set up some ratios for sine, cosine, and tangent.
Remembering the memory device SOHCAHTOA, sine will be the opposite over the hypotenuse. Cosine is equal to the adjacent side over the hypotenuse. And tangent is equal to the opposite side over the adjacent side. Starting with the equation we’re given, let’s plug in what we know. Our first term cos of 60 degrees: the cos of 60 degrees will be its adjacent side length over the hypotenuse. The adjacent side length is one, and the hypotenuse is two. Cos of 60 degrees is one-half.
Moving on to the next term, we’re trying to take sin of 30 degrees, which is the opposite side length over the hypotenuse. The opposite side length is one, and the hypotenuse is again two. Cos of 60 degrees times sin of 30 degrees is one-half times one-half. Sin of 60 degrees will be the opposite over the hypotenuse. If we’re using the angle of 60 degrees, its opposite is the square root of three and the hypotenuse is two. For tan of 60 degrees, we’ll need to take the opposite side length over the adjacent side length. The opposite side is the square root of three, and the adjacent side is one.
For the second term, we’ll need to multiply the square root of three over two times the square root of three over one. In the last term, we need to take the square of the tan of 30 degrees. Beginning with the angle 30 degrees, its opposite side is one and the adjacent side is the square root of three. At this point, we need to bring down all the operations. And we need to be particularly careful to make sure we bring down the square for the tan of 30 degrees.
Following the order of operations, we’ll multiply these fractions. One-half times one-half equals one-fourth. The square root of three over two times the square root of three over one, well the square root of three times itself just equals three and two times one equals two. We need to take the square of one over the square root of three, which should be one squared, over the square root of three squared. One squared is one. And the square root of three squared is just three.
Again bring down those operations. At this point, we need to add these three fractions together. But we can only add fractions with the common denominator. The least common multiple of four, two, and three would be 12. If we wanna rewrite one-fourth with the denominator of 12, we know that four times three equals 12. Therefore, we need to multiply the numerator by three as well. One-fourth equals three twelfths.
Bring down the subtraction. To get from two to 12, we multiply by six. If we multiply by six in the denominator, we need to multiply by six in the numerator. Three times six equals 18. Three-halves written as a fraction with the denominator of 12 is eighteen twelfths. Bring down the addition to rewrite one-third as the denominator of 12. Three times four equals 12. And one times four equals four.
Once we have a common denominator, we can add and subtract the numerators. We’ll have three minus 18 plus four twelfths. Three minus 18 is negative 15, plus four equals negative 11. The value of this expression is negative eleven twelfths.