# Trigonometry Application Worksheets

We often learn various things in school, and we wonder what the purpose is behind learning it. When it comes to trigonometry you do not need to look too far to realize why it is important. The applications of trigonometry in real life are robust and very impactful on the human experience. For over two thousand years we have been using it to create and navigate world maps to chart our ability to travel. Just about every form of construction relies on this form of math to plan and manufacturer just about everything. In this series of lessons and worksheets we will explore the application of trigonometry in real world situations and scenarios.

### Aligned Standard: High School Geometry - HSG-SRT.C.8

- Distance to the pole Step-by-step Lesson - See if you see the age slip here. It is fun to add those in word problems to see if kids catch the gaffe.
- Guided Lesson - Katie's kite string, height between trees, and the height of a ladder.
- Guided Lesson Explanation - The explanation show you that these problems are not that difficult.
- Practice Worksheet - I made sure to make a whole bunch of word problems for you. Bring on the elephants!
- Trigonometric Ratios 5 Pack - Find the legs of all these different, yet similar, triangles.
- Matching Worksheet - More word problems for you to tangle with.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Each sheet gets progressively more fun, at least to write.

- Homework 1 - Michael spots a cool tree. The tree creates a 600 angle, from the point where he is standing. The tree's height is 50 m. How far is Michael from the tree?
- Homework 2 - A 20 meter ladder is against a building. A 72 degree angle is formed underneath the ladder. Find the height of the building
- Homework 3 - Nancy is flying a kite. The kite is at a 60 degrees angle with the ground. The kite is 20 m high. Find the length (in meters) of the string that she used.

### Practice Worksheets

I really like the way that these guys came out.

- Practice 1 - An 80-foot pillar cast a shadow that is 40 feet long. Find the angle of elevation (degrees) of the sun at this point of time.
- Practice 2 - A hot air balloon flies 450 feet above the ground. Jenny sees the top of the balloon from 10 feet away the launching point. What is the angle of elevation (in degrees) that is made?
- Practice 3 - A 200 feet rope is nailed to a tree. The rope is held up on apole to form a right angle with the ground. The distance between nail and the base of tree is 100 feet. Find the angle made by rope with the tree?

### Math Skill Quizzes

An application problem can be found on the first quiz.

- Quiz 1 - Find the length of the hypotenuse of a right triangle, if the lengths of the other two sides are 13 inches and 12 inches.
- Quiz 2 - Find the length of one side of a right triangle, if the length of the hypotenuse is 11.6 inches and the length of the other side is 5.2 inches.
- Quiz 3 - In a ΔXYZ, XY = 65 and < X = 45 degrees. Find YZ?

### How is Trigonometry Used in Real Life?

The main use for this form of math in the daily life is to help us track different measurements. There are a number of applications that we use every day and do not even realize that they rely on trigonometry to operate such as traveling by airplane or playing console-based video games. It is a great tool to gauge distances and height on this planet and others. Here are just a few of the applications that we rely on from this form of math:

**Astronomy** - We calculate distance to stars and other planets via the use of trigonometric functions. Just about all space travel including the paths of rockets and space shuttles is modelled and navigated using this form of math. The concept of satellites was built and functions because of this. Which, by the way, is why and how GPS (Global Positioning System) functions. Thank trigonometric mathematicians next time you use Google Maps.

**Construction** - Just about everything that is being built from nothing is planned in every aspect using trigonometry. Surveyors explain every aspect of the land something will be built on measuring heights optical thanks to this form of math. If you take a look at blueprints that are designed by an architect, it looks like one huge proof. Even the contractors that use this receipt to bring the plans to life use trigonometry to make sure that their measures and cuts are precise.

**Video Games** - Have you played a video with some complex graphics that mimics real life characters? Yes, video games have come a long way to progress towards realty. Have you noticed that the mechanics of environments in which they are placed are similar to the movement of geometric shapes on a three-dimensional coordinate plane? That because most graphics engines used by game developers is built off a three dimensional trigonometric model.

### How does the Pythagorean Theorem relate to Trigonometric Ratios?

Pythagoras' theorem elaborates on the mathematical relationship between all three sides of a right-angled triangle. According to the theorem, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (Opposite and Adjacent). Its formula is written as: H^{2} = A^{2} + O^{2}. You can also prove the Pythagorean theorem with different trigonometric functions. By using sine, cosine, and tangent, you can find the missing sides and angles of the triangles. If you know the values of two sides of a triangle, you can find out the value of the third side and also the angle θ by the trigonometric functions.

In trigonometry, these functions can be written in the following ways as well. Sine Function: sin θ = (Opposite )/Hypotenuse, Cosine Function: cos θ = (Adjacent )/Hypotenuse, Tangent Function: tan θ = (Opposite )/Adjacent These worksheets and lessons show students how to manipulate trigonometric ratios and the Pythagorean theorem and use them as tools.