Remind students that the Right Triangle Table is useful. But what if the angle is not a multiple of 10 degrees? How can we move from estimating to calculating? Tell students that the answer is we can use trigonometric ratios.

Explain that the word “trigonometric” comes from the ancient roots of “trigon” (like “pentagon” or “hexagon”) and “metric” (like “measure”). So the word “trigonometric” comes from ancient words meaning “triangle measurement.”

Point out that the Right Triangle Table shows three trigonometric ratios:

• The ratio "adjacent leg divided by the hypotenuse" is called cosine.
• The ratio "opposite leg divided by the hypotenuse" is called sine.
• The ratio "opposite leg divided by the adjacent leg" is called tangent.

Tell students to label the columns of the Right Triangle Table with the corresponding trigonometric names.

Explain that cosine, sine, and tangent work like functions, where the input is the measure of an angle and the output is a ratio. Scientific calculators can display the ratio for any angle, even ones not included in the table.

For example, in the right triangle in this activity, one angle measures 50 degrees and the adjacent leg is 3 units long. If we want to know the length of the hypotenuse, we use cosine because it is the ratio of the length of the adjacent leg divided by the length of the hypotenuse. So we can write . Demonstrate how to use the technology available to solve this type of problem. Typing into the calculator and then substituting gives the new equation . (Note: If students don’t get 0.643, check the mode of the calculator. There is no need to discuss the meaning of radians now. Students will see radians later in this course.)

Arrange students in groups of 2. Ask students to compare their strategy with their partner’s and decide if they are both correct, even if they are different.

Students need not complete all the problems before the discussion.