Video: Using Trigonometric Values of Special Angles to Evaluate Trigonometric Expressions

Find the value of cos² 60° − 2 tan² 45° − 6 sin² 30°.

02:35

Video Transcript

Find the value of cos squared 60 degrees minus two tan squared 45 degrees minus six sin squared 30 degrees.

We’re going to begin by rewriting cos squared 60, two tan squared 45, and six sin squared 30 over so slightly differently. Cos squared 60 is simply cos of 60 degrees all squared. Similarly, two times tan squared 45 is two times tan of 45 all squared. Remember, we do the exponent before we do any multiplication. This also means that six sin squared 30 is the same as six times sin 30 all squared.

Now, we actually should know the values of cos of 60, tan of 45, and sin of 30 degrees by heart. But there is a table that can help us. We write sin, cos, and tan and 30, 45, and 60 degrees. We then write one, two, three and three, two, one. We then make the denominator of each of these two and we find the square root of the numerator. But since the square root of one is simply one, we leave sin of 30 and cos of 60 as one-half.

To find the values of tan, we divide the value for sin by the value of cos. And in fact, since their denominators are equal, this is just like dividing the numerators. So, tan of 30 is one divided by root three. Tan of 45 is root two divided by root two, which is simply one. And finally, tan of 60 is root three divided by one, which is root three.

We now use the table to find the values in our question. We begin by looking for cos 60, which is here. So, cos 60 squared is one-half squared. Tan of 45 is here; it’s one. So, two tan of 45 squared is two times one squared. And sin of 30 is one-half. So, six sin of 30 all squared is six times a half squared. Our expression then becomes a half squared minus two times one squared minus six times a half squared.

To square a fraction, we square the numerator and the denominator. So, a half squared is one-quarter. Two times one squared is just two times one, which is two. And then our final term is negative six times a quarter. Six is the same as six over one. And when we multiply two fractions, we multiply the numerators and then multiply their denominators. So, six times one-quarter is six-quarters. And of course, we can simplify this but I haven’t chosen to just yet. And that’s because we can create a common denominator to simplify.

Two is the same as two over one. And if we multiply both the numerator and denominator by four, we get eight-quarters. So, our expression becomes a quarter minus eight-quarters minus six-quarters. Once the denominators are equal, we can add or subtract the numerators. And when we do, we find that cos squared 60 minus two tan squared 45 minus six sin squared 30 is negative 13 over four.

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