Find 𝑥 to two decimal places.
In this question, we’re asked to find the value of 𝑥 to two decimal places. And we can see that 𝑥 is the length in a right triangle. And we can see we’re given one of the non-right angles of this right triangle and one of the lengths of this right triangle. And we can recall to find an unknown length in a right triangle given an angle and another length, we can use right triangle trigonometry. And to do this, we first need to label the sides of the right triangle. We need to label them based on their position relative to the angle of 55 degrees.
First, we can see the side labeled 𝑥 is opposite the angle 55 degrees. So this is the opposite side. Next, the longest side in the right triangle, that’s the one opposite the right angle, is called the hypotenuse. In this case, this has a length of 10. Finally, the side next to the angle of 55 degrees, which is not the hypotenuse, is called the adjacent side. This is because it’s adjacent to our known angle.
We now need to determine which of our trigonometric ratios we need to use to find the value of 𝑥. And to do this, we can recall the acronym SOH CAH TOA. In our right triangle, we want to determine the length of the opposite side and we know the length of the hypotenuse. In other words, we want to determine O, and we know the value of H. Therefore, the acronym SOHCAHTOA tells us we should use S, which represents the sine function. We can recall the sine function tells us the ratio of the length of an opposite side and hypotenuse in a right triangle. In other words, if 𝜃 is an angle in a right triangle, then the length of the opposite side divided by the length of the hypotenuse in this right triangle is sin 𝜃.
In our case, our value of 𝜃 is 55 degrees, the length of the opposite side is 𝑥, and the length of the hypotenuse is 10. Therefore, by applying right triangle trigonometry, we get the sin of 55 degrees is equal to 𝑥 divided by 10. We can then solve for the value of 𝑥 by multiplying through by 10. We get that 𝑥 is equal to 10 times the sin of 55 degrees. And we can evaluate this by using our calculator, where we need to be sure our calculator is set to degrees mode. If we input this expression into our calculator, we get that 𝑥 is equal to 8.191, and this expansion continues.
But remember, the question wants us to give our answer to two decimal places. So we look at the third decimal digit, which is one. Since this is less than five, we need to round down, which gives us 8.19, which is our final answer. Therefore, we were able to show for 𝑥 given in the diagram, to two decimal places, 𝑥 is 8.19.