Video: Understanding the Tangent Ratio

In the given figure, ๐‘šโˆ ๐ต๐ด๐ถ = 90ยฐ and ๐ด๐ท โŠฅ ๐ต๐ถ. What is ๐ด๐ถ tan ๐œƒ?

02:16

Video Transcript

In the given figure, the measure of angle ๐ต๐ด๐ถ is equal to 90 degrees and the line ๐ด๐ท is perpendicular to ๐ต๐ถ. What is ๐ด๐ถ tan ๐œƒ?

So in this question, weโ€™ve got a right-angle triangle cause weโ€™re told that the measure of angle ๐ต๐ด๐ถ is equal to 90 degrees. So thatโ€™s a right angle. So therefore, because weโ€™ve got a right-angle triangle, we know that we can have a look at the trigonometric ratios. And when weโ€™re dealing with the trigonometric ratios, we have a memory aid, which is SOHCAHTOA. And what this does is it tells us how to work out sin ๐œƒ, cos ๐œƒ, and tan of ๐œƒ.

In this question, weโ€™re looking for ๐ด๐ถ tan ๐œƒ. Well therefore, we know that weโ€™re gonna be interested in TOA because thatโ€™s the one that deals with the tangent ratio. And what TOA tells us that we said is that tan ๐œƒ is equal to the opposite divided by the adjacent. So what do we do now? Well, the next step is to label our triangle.

Well, the first side that weโ€™re gonna label is our hypotenuse. And thatโ€™s because this is the side thatโ€™s opposite the right angle. And itโ€™s also the longest side of our triangle. And the triangle weโ€™re interested in is the triangle ๐ด๐ต๐ถ. Now, the next side weโ€™re gonna label is the opposite because this is the side opposite the angle, which is ๐œƒ. And the final side is the adjacent. And this is the side thatโ€™s next to the angle ๐œƒ and also between the angle ๐œƒ and the right angle.

So in this question, weโ€™re interested in the opposite and the adjacent because weโ€™re dealing with tan ๐œƒ. So therefore, if we substitute these into our formula for tan ๐œƒ, we can see that tan ๐œƒ is equal to ๐ด๐ต, the opposite, divided by ๐ด๐ถ, which is the adjacent. And then, what we can do is multiply each side of the equation by ๐ด๐ถ to remove it from the denominator so to get rid of the fraction.

And when we do that, we get ๐ด๐ถ tan ๐œƒ is equal to ๐ด๐ต. So that solved the problem because what we were looking for is what ๐ด๐ถ tan ๐œƒ was. And ๐ด๐ถ tan ๐œƒ, as weโ€™ve already said, is ๐ด๐ต.

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