Problem Solving with Trigonometric Ratios

We have just recently completed our unit on trigonometry and had our unit test this week. As we neared the end of the unit we had learned most of the important skills of trigonometry. This meant that we were now going to be putting what we learned in class to work by interpreting word problems and real life problems and solving them with the use of trigonometric ratios. This was a good test of our knowledge and how well we really understood the topic since word problems will really make you think. I believe that learning problem solving methods is very important as trigonometry problems are very likely to show up in daily life, especially when measuring things.

I will now show an example of a world problem where trigonometry will be needed and how to solve it. Here is the first question:

Step 1 – drawing the diagram: first we are going to identify what sides we are working with and what formulas need to be used. First of all it tells us that the Eiffel Towers is 1063 ft high, so this is going to be the height of our triangle. Then it tells us that the angle of elevation is 34.1° which is the angle looking up from the ground. Then the question is asking for the length of the shadow which will be the horizontal side of the triangle. From the information that we have been given we can determine that the side length needs to be found for this question.

Step 2 – writing the acronym: this is a very important step when dealing with trigonometry. Always remember to write SOH CAH TOA at the beginning of a question to help yourself out.

Step 3 – labelling the sides: like with all the other trigonometry questions the sides need to be labelled to determine which function is going to be used in the question. First we can label the hypotenuse which is the longest side and easiest to figure out. Next label the opposite side which is opposite of the reference angle, in this case 34.1°. Next the final side left over will be the adjacent side.

Step 4 – determining the function: now that we have our sides labelled we can determine the function by recognizing which sides we are working with. The first side is going to be the adjacent side that “x” is on (the length of the shadow). The next side is the side that we know, which in this case is the opposite side (the height of the triangle or Eiffel Tower). And since we are working with the adjacent and opposite side, the tangent function needs to be used.

Step 5 – writing the equation: now that we have determined the tangent function will be used we need to write our equation. This will simply be the tangent of the angle = the opposite side divided by the adjacent side.

Step 6 – solving the equation: now all that is left is solving the equation. First we are going to multiply each side by the denominator of the fraction which happens to be “x”. When “x” is in the denominator it makes things a bit trickier, but not impossible to do. Once each side is multiplied by “x”, 1063 will separated and all that is left is to solve. In order to do this, each side needs to be divided by the tangent of 34.1.

Note: I have rounded to the nearest whole number in this case.

 

Example 2: here is the second example question we are going to be working with.

Step 1 – splitting the triangle: in this question the goal is to find the length of side “PQ”. This may seem odd at first because this triangle is not a right angle triangle, however this issue can be fixed. As you can see I have split the triangle into two right angles which will be solved exactly the same way as all the previous questions we have done.

Step 2 – writing the acronym: once again we are going to write out one of the most important parts of the equation to help us determine and remember which function needs to be used.

Step 3 – labelling the sides: since we have two triangles we are going to have to label the sides for both triangles which requires the same process. After this is done we will be able to determine the function that needs to be used. First we can label the hypotenuse which is the longest side and easiest to figure out. Next label the opposite side which is opposite of the reference angle. Next, the final side left over will be the adjacent side.

Step 4 – determining the function: now that we have our sides labelled we can determine the function by recognizing which sides we are working with. For the first triangle we need to find “x” which is going to be the adjacent side. The next side is the side we know which is 7.4 and is also the opposite side. So for this triangle since we are working with the adjacent and opposite side, the tangent function needs to be used. For the second triangle we need to find “y” which is going to be the adjacent side. The next side is the side we know which is 7.4 and is also the opposite side. So for this triangle just like the first we are working with the adjacent and opposite side so the tangent function needs to be used.

Step 5 – writing the equation: now that we have determined the tangent function needs to be used for both triangles an equation needs to be written out. This will simply be the tangent of the angle = the opposite side divided by the adjacent side.

Step 6 – solving the equation: now all that is left is solving the equation. For the first triangle we are going to multiply each side by the denominator of the fraction which happens to be “x”. When “x” is in the denominator it makes things a bit trickier, but not impossible to do. Once each side is multiplied by “x”, 7.4 will separated and all that is left is to solve. In order to do this, each side needs to be divided by the tangent of 41. For the second triangle the same things needs to be done. Each side needs to be multiplied by “y”, which will cause 7.4 to be separated and all that is left is to solve. To solve, each side needs to be divided by the tangent of 38.

Note: I have rounded to the nearest tenth in this question.

Step 7 – adding the sides together: since we split the two triangles, we got two separate measurements which only equal to a part of side “PQ”. So in order to get the final answer we must add our two answers together. After this is done we have found the length of side “PQ”.

 

Summary: 

These are two examples of problem solving using trigonometry. I believe that this is such a valuable skill as triangles are everywhere in our daily lives even though they may not always be obvious. If trigonometry is used, very large measurements can be found with ease and accuracy every time which can be very useful and much safer than physically measuring certain things.