Lesson 8

Sine and Cosine in the Same Right Triangle

Lesson Narrative

In a previous lesson, students built a right triangle table. Each group had a pair of complementary angles, so they began to make conjectures about cosine and sine of complementary angles before they learned the terms cosine or sine. In this lesson, students do some calculations to remind them of their previous conjectures and then prove \(\sin(\theta)=\cos(90-\theta)\).

Throughout this lesson there is a focus on precision of language. The warm-up prompts students to compare four triangles. The Which One Doesn't Belong? routine gives students a reason to use language precisely (MP6). The following activity asks students to explain how they got the same answers as their partner despite being assigned different triangles (the pairs of triangles were congruent but had different angles provided). In the final activity students write a draft of a proof, work with their group to refine the group proof, and then have a whole class discussion on how to clearly communicate ideas using words and diagrams.

Learning Goals

Teacher Facing

  • Explain the relationship between the cosine and sine of complementary angles (using words and other representations).

Student Facing

  • Let’s connect cosine and sine.

Learning Targets

Student Facing

  • I can explain why $\sin(\theta)=\cos(90-\theta)$.

CCSS Standards

Building On

Addressing

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.