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Trigonometric Ratios

HSG-SRT.C.7
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Aligned To Common Core Standard:

Trigonometry - HSG-SRT.C.7

What are Cos and Sin Trigonometric Ratios? Trigonometric ratios apply only to right-angle triangles. Right-angled triangles have the specialty of having one 90 degrees angle while the other two angles are less than 90 degrees. In these triangles each side has a specific name: Hypotenuse - It is the largest side of the right-angle triangle. It lies opposite to the right angle of the triangle. Base - The side on which the right-angle triangle stands is called as a base. The two sides, except hypotenuse, can be selected as a base to perform calculations. Perpendicular - the side perpendicular to the base is known as the perpendicular of the right-angle triangle. Trigonometric Ratios - Trigonometric ratios are used to find all measures of a triangle. They are used to find the ratios of the sides of the right-angled triangle. Trigonometric ratios are of three types; sine, cosine and tangent. For now, we will stick to cosine and sine ratios. We can easily find any unknown side of the right-angle triangle by using trigonometric ratios if we know: One length and One angle, other than the right angle. When using trigonometric ratios, fixing perpendicular and base of the right-angle triangle can sometimes be confusing. Therefore, to make things less confusing, we can also name as: Adjacent/base: -The side next to the given angle. Opposite/ Perpendicular -The side opposite to the given angle. The longest side is considered as the hypotenuse. Cosine and Sine - Sine is defined as the ratio of the opposite side to the hypotenuse: Sine Θ = (Opposite )/Hypotenuse While cosine is defined the ratio of the adjacent side to the hypotenuse: Cosine Θ = (Adjacent )/Hypotenuse

Printable Worksheets And Lessons


  • Cos and Sin Step-by-step Lesson - Again we go with baby steps on this one. Very fundamental in the approach that we take.

  • Guided Lesson - Find the height of Jacob's building. How tall is that tree? How far is Irvin from his sticker?

  • Guided Lesson Explanation - Man, I need to come up with more interesting questions. I mean; why would ever need to calculate how far you are from a sticker?

  • Practice Worksheet - This sheet gets back to basics and shies away from the word problems that we just introduced.

  • Matching Worksheet - This one is very deceiving. The problems are tougher than they appear.

  • Find the Tangent of One Point - By about problem number 6 you should have this skill mastered.



Homework Sheets

Each sheets progressively steps up in difficulty.

  • Homework 1 - Tangent Ratio: for any acute angle θ of a right triangle.
  • Homework 2 - A 30 feet ladder rests against a wall. The base of the ladder is 10 feet from the wall. What is the height of the wall?
  • Homework 3 - Pearl went to the market. She saw a ladder there. She stays at the point X, where shadow of the ladder ends. The distance from that point, to the top of the ladder point Y is 40 feet. The measured height of the ladder is 55 feet. What is the distance of point W to the point X of wall?



Practice Worksheets

Each sheet has two basic ratio questions and a word problem.

  • Practice 1 - William saw a communication antenna outside his building. The shadow of the communication antenna is 12 meters. He then sees the top of the communication antenna. The slope of the communication antenna is 16 meters long. What is the height of the communication antenna?
  • Practice 2 - One day Angel went to the park. She was playing around a tree. She stays at the point U, where shadow of the tree ends. The distance from that point, to the top of the tree point W is 80 meter. The measured height of the tree is 56 meters. What is the distance of point U to bottom point V of the tree?
  • Practice 3 - Nile loves his remote car. His mother put the remote car at the point T, which is top of the cupboard. The height of the cupboard is 25 feet. Nile looks from his knees at the point S and saw the remote car. His position makes a 40 degrees angle. Calculate the distance from point S to point T.



Math Skill Quizzes

The quizzes are all skill focused and not application based.

  • Quiz 1 - Write each trigonometric ratio. Even though there are only four triangles, each triangle has four problems attached to it.
  • Quiz 2 - We working on using tangent measures to find missing values.
  • Quiz 3 - Find the missing side using your understanding of tangent ratios.


How Does This Skill Apply to The Real World?

Many times, over our math learning career we will study a topic and learn a new skill but have no idea how it will ever help us in life. There are so many different applications of Sine, Cosine, and Tangent ratios that we do not even know where to start. If you have something in front of you that possesses a right-angled triangle, you can apply this to find a missing side or angle. If you wanted to know how tall a building is, you could apply this. If you wanted to buy a ladder that would get you to the top of your house, this math could be used to figure out how big of a ladder you would need. You could use angle measures to determine how far away from something you are. This can be applied on Earth and outside of our world in space as well. These measures are constantly being calculated by instruments within airplanes to provide the pilot with an understanding of their environment. We can also use this under the ground to locate historical items that have been buried. It can also be used in the world’s oceans to determine measures of structure that reside on the floor of the ocean. Of course you have to take into account the properties of salt water and soil, but it can be done pretty accurately through the use of this type of math.