## Trigonometric Ratios and the Pythagorean Theorem

#### Aligned To Common Core Standard:

**Trigonometry** - HSG-SRT.C.8

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.

### Printable Worksheets And Lessons

- 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.

#### 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?