Video Transcript
Find the values of 𝑥 and 𝑦 giving the answer to three decimal places.
In this question, we need to determine the values of two unknowns 𝑥 and 𝑦, and we need to give our answer to three decimal places. To answer this question, we first notice that the values of 𝑥 and 𝑦 represent side lengths in a right triangle. And in this right triangle, we’re given one of the non-right angles and one of the other side lengths. Therefore, we can determine the values of 𝑥 and 𝑦 by using right triangle trigonometry.
To do this, we first need to label the size of this right triangle. We should always start with the hypotenuse. It’s the longest side in the right triangle, which is the one opposite the right angle. So, in this case, the hypotenuse has length 𝑦. Next, we need to label the other two sides based on their position relative to the angle that we know. That’s 47 degrees. First, we can notice the side of length 28 centimeters is opposite the angle of 47 degrees. So, we’ll label this side as the opposite side. Finally, the last side of length 𝑥 is next to our angle of 47 degrees, so we could say that it’s adjacent to this angle. We’ll label this side the adjacent side.
We can now recall that we can determine the values of 𝑥 and 𝑦 by using the trigonometric ratios. To help us determine which trigonometric ratios we need to use, we’ll recall the following acronym: SOH CAH TOA. We’ll need to apply this twice to determine the values of 𝑥 and 𝑦 separately. Let’s start by determining the value of 𝑥. This means we need to start by using our acronym to determine which trigonometric ratio will help us find the value of 𝑥.
We can see that 𝑥 is the adjacent side to our angle. And we already know the value of the opposite side to our angle. And we can see in the acronym if we know the opposite side to the angle and the adjacent side to the angle, then we need to use the tangent function. We know if 𝜃 is an angle in a right triangle, then the tangent of angle 𝜃 tells us the ratio of the length of the opposite side to angle 𝜃 divided by the length of the adjacent side to angle 𝜃. In our right triangle, we want to use the angle 𝜃 is 47 degrees. Then the adjacent side has length 𝑥, and the opposite side has length 28 centimeters.
Substituting these values in, we get the tangent of 47 degrees is equal to 28 divided by 𝑥. We need to solve this equation for 𝑥. First, we’ll multiply both sides of our equation through by 𝑥. This then gives us that 𝑥 times the tangent of 47 degrees is equal to 28. And now, we can solve for 𝑥 by dividing both sides of the equation through by tangent of 47 degrees. We get that 𝑥 is equal to 28 divided by the tan of 47 degrees. And now, by remembering the side lengths of this triangle are measured in centimeters and by using our calculator, where we make sure our calculator is set to degrees mode, we can find the value of 𝑥. It’s 26.1104 and this expansion continues centimeters.
Finally, the question wants us to give our values of 𝑥 and 𝑦 to three decimal places. So, we look at the fourth decimal place, which is four, which we know is less than five. So, we need to round this down. Therefore, to three decimal places, 𝑥 is 26.110 centimeters.
We can follow the exact same process to find the value of 𝑦. Once again, we’ll use our acronym of SOH CAH TOA to determine which trigonometric ratio we need to use. In the diagram, we know the length of the opposite side, and we want to determine 𝑦, which is the length of the hypotenuse. Therefore, we want the trigonometric ratio relating the opposite side and the hypotenuse. That’s the sine function. We can then recall if 𝜃 is an angle in a right triangle, then sin 𝜃 is equal to the length of the side opposite angle 𝜃 divided by the length of the hypotenuse.
We can then substitute in our values from the diagram. We get the sin of 47 degrees is equal to 28 divided by 𝑦. Now, all we need to do is solve this equation for 𝑦. We multiply both sides of the equation through by 𝑦 and then divide through by the sin of 47 degrees. We get 𝑦 is equal to 28 divided by sin of 47 degrees. Now, we evaluate this expression by using our calculator. We get that 𝑦 is 38.2851 and this expansion continues centimeters. Finally, we need to round this to three decimal places. The fourth decimal digit is one, which is less than five. So, we need to round this value down. This gives us that 𝑦 is 38.285 centimeters to three decimal places.
Therefore, by using right triangle trigonometry, we were able to find the values of 𝑥 and 𝑦 in the diagram to three decimal places. We saw that 𝑥 was 26.110 centimeters and 𝑦 was 38.285 centimeters.
