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Video: Finding the Unknown Length in a Right Triangle Using Trigonometry Where the Unknown Is on the Bottom of the Fraction

Kathryn Kingham

Find 𝑥 to two decimal places.

03:34

Video Transcript

Find 𝑥 to two decimal places.

What are some things we know about this problem simply by looking at the picture? We know that we’re dealing with a triangle, specifically this is a right triangle. We know that because of this symbol, we know that 𝑥 is our hypotenuse or our longest side. We know that because the 𝑥 is the side opposite the right angle. We have one angle that’s 20 degrees. We also know that we have the side length opposite our 20 degrees.

But we also need some information that we can’t get from the picture. We have to remember the trigonometry functions. Here are the three main ones we might use. Sine is when you’re working with the opposite angle over the hypotenuse. Cosine equals the adjacent angle over the hypotenuse. Tangent would be the opposite angle over the adjacent angle.

We have an angle. We’re missing the hypotenuse. And we’re working with the side that’s opposite the angle that we’ve been given. With the information we have, it’s best for us to use the sign because we’ve been given a side length opposite the angle we know, and we’re looking for the hypotenuse.

When we plug in what we know, we say sign of 20 degrees equals 12 over 𝑥. But we’re solving for 𝑥 here, so we need to get 𝑥 by itself. We can multiply both sides of this equation by 𝑥 over one. On the right side, the 𝑥𝑠 cancel out leaving us with 12. On the right side, we now have 𝑥 times sin of 20 degrees. We’ll need to get rid of sin of 20 degrees and move it to the other side of the equation. To do that, we’ll divide both sides of our equation by sin of 20 degrees. Sin of 20 degrees divided by sin of 20 degrees cancels out.

And now we have 𝑥 equal to 12 divided by sin of 20 degrees. You can plug this part into the calculator. 12 divided by sin of 20. The calculator tells us that 𝑥 would be equal to 35.0856528. And our question would like to know 𝑥 accurate to two decimal places. Taking a closer look here, to round, we know that the five in the thousands place makes the eight in the hundreds place round up, which tells us that our 𝑥-value, our hypotenuse for this triangle, is approximately 35.09 units.