Trigonometry Word Problems and How to Solve Them

trigonometry word problemsMathematics and trigonometry have become essential for students who want to pass their GCSE exams to secure their future careers. Word problems are an essential part of passing mathematics or trigonometry. Understanding how to translate word problems into mathematical solutions is an essential skill for students to master…and easy to learn if you learn it the right way! Trigonometry word problems include problems relating to radians and degrees, circles, word problems involving trigonometric functions, and word problems involving identities.

This blog will show you how to work with trigonometry word problems. If you want to learn how to conquer trigonometry word problems, then sign up for the Trigonometry: Degrees and Radians course. This course contains over twenty two lectures that will teach you how degrees, revolutions and radians are defined, how to convert between degrees and radians, how to draw angles in a standard position, how to solve problems involving large angles and how to find the exact values of the six trigonometric ratios for a given triangle.

Solving Trigonometry Word Problems

To translate trigonometry word problems into mathematical equations and solutions, you need to have a good understanding of the concepts within trigonometry, as well as the definitions of these concepts. Trigonometry is often expressed as an image representing the angles, circles and other trigonometric concepts involved. It is therefore best to translate your word problem into a picture that represents the word problem. Use the words to visualize and draw the image involved to represent the problem. We will use this approach for the trigonometry problem examples included in this tutorial.

The Flagpole Trigonometry Word Problem Example

One of the most common word problems you will come across in trigonometry is the flagpole example. In this type of word problem, you are generally given two values for calculation purposes and you are asked to find the missing information. A good example of this type of word problem is:

A flagpole is 18 feet high. The flagpole casts a shadow of 24 feet along the ground. Calculate the distance between the end of the shadow and the top of the flag pole and calculate the angle between the shadow and the line representing the distance.

trigonometry word problemsTo solve this type of trigonometry word problem I also begin by drawing the word problem. Your drawings do not have to be perfect. As long as they represent the essence of the word problem and you can visualize the problem then the drawing is good enough. Here is my drawing of the above problem:



Now you need to add all of the information you know about the problem to the image you have drawn. We know that there is a ninety degree angle between the ground and the flagpole. We also know the angle between the shadow and the distance is equal to sin. And we also know that we can use Pythagoras’ theorem for this particular triangle. We also know that a lot of these types of triangles confirm to the 3,4,5 scale. So let’s add that information to our diagram:


We can see from the diagrammatic representation we have drawn above that the 3,4,5 rule will apply here since 6×3 = 18 which is the height of the flagpole, 6 x 4 = 24 which is the length of the shadow. So now all we need to do is work out the 5 portion: 6 x 5 = 30.

Now that we have the distance of the shadow, we can use simple trigonometry to calculate the sin of the angle:

sin = 18 / 30

For more trigonometry word problems, sign up for the Trigonometry: Trigonometric Functions II course. This course offers over twenty lectures that include word problems to calculate functions of angles, and other simple applications of trigonometry such as pendulum, wind turbine, helicopter and ferris wheel word problems.

The Airplane Trigonometry Word Problem Example

Another common trigonometric example is one where you are given an angle of elevation and distance to work with and you are asked to find various elements of the equation. Here is an example of the airplane example. This example can relate to anything that is flying – a bird, a plane, a rocket, a balloon.

Anne sees a rocket at an angle of elevation of 11 degrees. If the rocket launch pad is 5 miles away from Anne, then how high is the rocket?

Once again, we start this problem by drawing a diagram that represents the word problem. Remember – you are not being marked for artwork, you are just drawing the problem to be able to see what information you already have. This is my drawing (Please ignore my artistic talent):


From the diagram we can see that we know the angle is eleven degrees. We also know we have a right angle between the ground and the rocket so we are dealing with the Pythagorean Theorem again. So we are essentially dealing with the same problem as the flagpole problem now. We will call the side of the triangle that represents the height of the rocket x for our equation. We now have x, 5 and 11 degrees.

We also know that tan = opposite / adjacent

So tan 11 degrees = x / 5

Or 5 tan 11 degrees = x

Now you can plug 5 times tan 11 degrees into your calculator for the answer.

For more examples of how to solve trigonometric word problems, sign up for the Trigonometry: The Unit Circle course.

The Truck Trigonometry Word Problem Example

Our next word example takes our trigonometric knowledge one step further than the above two examples. In this type of word problem students are given a vehicle like a car or a boat or a train and they are also given two angles of elevation and the students are asked to calculate how far the vehicle has travelled. Here is an example of this type of word problem:

A helicopter is hovering 800 feet above a road. A truck driver observes the helicopter at a twenty degree angle. Twenty five seconds later the truck driver notices the angle of the helicopter is now at sixty degrees. How fast is the truck moving? Round your answer to the nearest foot.

Once again to answer this word problem, we first need to draw a diagram to represent what we know about the problem. This is my diagram for the above problem:


My artwork may not win any art awards, but it does show us the information we currently know. We know that the helicopter forms a right angle with the ground. We know the two angles of the two triangles and we know the helicopter is 800 feet above the ground. The distance the truck travels in 20 meters is the difference between the 65 degree triangle and the 20 degree triangle. And we can actually work out these distances from the helicopter. You can actually break down this diagram further into two separate right angle triangles:


According to our trigonometry functions the tangent of the angle is equal to the opposite over the adjacent. So:

Tan 65 = 800 / A

To find A, multiply by A on both sides:

A tan 65 = 800


A = 800/ tan 65

Putting those numbers into your calculator you will find A is equal to 374ft.

According to the above, we can use tan to work out the distance of B using the same steps:

Tan 20 = 800 / B

To find B, multiply by B on both sides of the equation.

B = 800 / tan 20

Using your calculator the distance from the helicopter to the truck (B) is 2197ft.

Now we can work out how far the truck travelled in twenty seconds:

B – A = 1823 ft

So if the truck traveled 1823 feet in twenty five seconds then the distance travelled per second is:

1823/25 = 73 feet per second

Now all you need to do is convert the answer to miles per hour.

Start Practicing Your Trigonometry Word Problems Today.

Essentially, irrespective of the trigonometry problems you need to solve, if you start by representing the problem as a diagram, you will quickly be able to see what information the problem already gives you and what the problem is asking you to solve. As long as you know the definitions of the trigonometric elements and know the equations, you should have no problem solving your trigonometry word problems.

If you are keen to learn more trigonometry to master this form of mathematics, then sign up for the Master Analytic Trigonometry course today. This course offers over eight lectures that will take you through the step by step processes necessary to simplify trigonometric equations, the steps you need to verify equations and the steps you need to solve Trigonometric equations.