Mathematics Common Core Standards: Similarity, Right Triangles, and Trigonometry
The Common Core Standards in this group focus primarily on triangles. Students discover the concept of similarity and scale and explore how these concepts apply to real-world situations. The unique nature of triangles also reveals certain mathematical truths about ratios that are very useful in solving a host of problems.
In geometry, similarity refers to triangles that have exactly the same shape but differ in size. Similarity differs from congruence, which describes triangles of identical size and shape. The use of similarity to represent larger objects is commonplace in fields such as engineering and architecture, when someone needs to accurately represent the size of certain objects on a smaller scale.
Students work extensively with right triangles (triangles with one 90-degree angle). Right triangles are unique in that you can find the length of any one side of the triangle if you know the lengths of the other two sides.
According to the Pythagorean theorem, “the square of the hypotenuse is equal to the sum of the squares of the other two sides,” which can be expressed as c2 = a2 + b2, where c is the hypotenuse (the longest side of the right triangle), and a and b are the other two (shorter) sides.
Trigonometry deals with the study and use of ratios involving triangle sides and angles. Students use the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) to solve for missing parts (including a missing side or angle) of a right triangle:
Using trigonometric ratios, you can determine the length of a side of a right triangle without knowing the lengths of the other two sides. You can determine the unknown length of a side given the length of one side and the angle next to it. Take a look for a visual representation of the opposite, adjacent, and hypotenuse sides when labeled for the purposes of practicing using these ratios.
|sine||opposite||angle & hypotenuse|
|hypotenuse||angle & opposite|
|angle||opposite and hypotenuse|
|cosine||hypotenuse||angle & adjacent|
|adjacent||angle & hypotenuse|
|angle||adjacent & opposite|
|tangent||opposite||angle & adjacent|
|adjacent||angle & opposite|
|angle||opposite & adjacent|
Challenge your child to choose the correct ratio and find the missing parts.
Draw right triangles in various sizes and dimensions that have different missing parts and have your child use the trigonometric ratios of sine, cosine, and tangent to find the missing angle or side of each triangle. You see some examples grouped by cosine, sine, and tangent, but don’t tell your child which ratio to use; challenge her to figure it out.
For example, to find x (opposite side) of the triangle in the upper left, knowing the angle is 19° and the hypotenuse is 4, you use the sine function:
Being able to accurately determine the missing side or angle of a right triangle is important because of the frequency of right triangles in the real world, including square objects that are divided into two right triangles and angles of elevation.
Trigonometric ratios are even more useful for finding missing sides and angles of non-right (oblique) triangles — triangles that don’t have a 90-degree angle — because the Pythagorean theorem works to find only missing sides of right triangles.
You can use the sine (sin), cosine (cos), and tangent (tan) buttons on your child’s calculator when working problems that require the use of any of these trigonometric ratios.