Video: Using Right Triangle Trigonometry to Solve Word Problems

A builder leans a 5-foot plank of wood against a vertical wall. If one end of the plank is 3 feet from the bottom of the wall, what is the angle between the plank and the wall?

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Video Transcript

A builder leans a five-foot plank of wood against a vertical wall. If one end of the plank is three feet from the bottom of the wall, what is the angle between the plank and the wall?

We have a vertical wall. A builder leans a five-foot plank of wood against this wall. The end of the plank is three feet from the bottom of the wall. What is the angle between the plank and the wall?

Because we know that the wall is vertical, the angle created here would be a right angle. And that means we can use our right angle trig functions to solve this problem.

We have an unknown angle, the opposite side length, and the hypotenuse. If you aren’t sure why this is the hypotenuse, note that the hypotenuse is always the side length opposite the right angle.

Okay, back to this relationship, the opposite over the hypotenuse is the sin of 𝜃. It is the sine relationship. The sine of our missing angle is equal to the opposite side length, three, over the hypotenuse side length of five. In order to isolate 𝜃, we need to take the sine inverse of the sin of 𝜃. And if we take the sine inverse of the left side of the equation, we need to take the sine inverse of the right side of our equation. Sine inverse of sin 𝜃 cancels out and leaves us with our 𝜃. Sine inverse of three over five equals about 36.8699 degrees.

Just a note, if your calculator told you that the sine inverse was 0.6435 continuing, then your calculator is set to radians. And in this problem, we’re trying to solve in degrees. You’ll need to change your settings so that your calculator is operating in degrees.

Now that we know what the angle measure is, we can round it to the nearest hundredth. To round to the nearest hundredth, we’ll look at the thousandths digit, which is a nine. That tells us we round our six up to a seven, and everything to the left of that digits stays the same. The angle between the plank and the wall is 36 and 87 hundredths degrees.

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