Lesson 7

Applying Ratios in Right Triangles

Lesson Narrative

In this lesson students deepen their understanding of trigonometry by completing an info gap. The info gap structure requires students to make sense of problems by determining what information is necessary and then asking for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6). During this process students will reinforce the idea that trigonometry is based on right triangles. So far students have only encountered trigonometry with triangles where the angle is given, so at this point they need one side length and one acute angle measure to calculate the remaining side lengths and angle measures. Later in this unit they will learn how to compute angle measures of right triangles given only side lengths.

Students then have the opportunity to apply this knowledge to some indirect measurement problems. During the discussion they consider rounding error again, this time by comparing with an exact value. The effect of rounding is much more pronounced with large numbers, so this discussion may convince students who were skeptical in previous lessons.

Learning Goals

Teacher Facing

  • Ask for information needed to calculate side lengths in right triangles (orally).
  • Determine the heights of objects using trigonometry (using words and other representations).

Student Facing

  • Let’s solve problems by using right triangles and trigonometry.

Learning Targets

Student Facing

  • I can use cosine, sine, and tangent to find the height of an object.

CCSS Standards

Glossary Entries

  • cosine

    The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse. In the diagram, \(\cos(x)=\frac{b}{c}\).

  • sine

    The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse. In the diagram, \(\sin(x) = \frac{a}{c}.\)

  • tangent

    The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg. In the diagram, \(\tan(x) = \frac{a}{b}.\)

  • trigonometric ratio

    Sine, cosine, and tangent are called trigonometric ratios.