Question Video: Evaluating Trigonometric Expressions Using Trigonometric Values of Special Angles | Nagwa Question Video: Evaluating Trigonometric Expressions Using Trigonometric Values of Special Angles | Nagwa

Question Video: Evaluating Trigonometric Expressions Using Trigonometric Values of Special Angles Mathematics • Third Year of Preparatory School

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Find the value of tan 60°(1 − tan² 45°)/sin 30° cos 60° without using a calculator.

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Video Transcript

Find the value of tan of 60 times one minus tan of 45 squared over sin of 30 times cos of 60 without using a calculator.

In this question, there are four trigonometric values of special angles. These are ones that we need to know by heart. And luckily, there’s a nice way to help us recall these. We’re going to draw a table. And on the left-hand side, we write sin 𝜃, cos 𝜃, tan 𝜃. And along the top, we write 30 degrees, 45, and 60. Sin 𝜃 and cos 𝜃 are fairly straightforward. We write one, two, three and then three, two, one. We then write these all as a fraction with a denominator of two. And the final step is to take the square root of all of the numerators.

Now, of course, the square root of one is simply one. So we really don’t need to write the square root here. And we find the value of sin of 30 and cos of 60 to be one-half. Tan of 𝜃 is not so straightforward. We divide the value for sin of 𝜃 by the value of cos of 𝜃 to get the value of tan 𝜃. So tan 𝜃 would be a half divided by root three over two. But since the denominator is the same, we simply divide the numerators. We get one divided by the square root of three. Then tan of 45 is root two over root two which is one. And tan of 60 is root three divided by one, which is root three.

We’re going to substitute all the values we know into our expression. Tan of 60 is the square root of three. Tan squared of 45 is one squared. And that’s all over sin of 30 times cos of 60, so it’s a half times a half. This becomes Root three times zero all over one-quarter. And then, that’s zero over a quarter. And we find the value of tan 60 times one minus tan squared of 45 over sin of 30 times cos of 60 to be zero.

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