Exploring Trigonometric Ratios

G-SRT.3.6

In this activity, students will [b]relate the definitions of trigonometric ratios to a property of similar triangles. [/b]

Triangle ADE

Note that dragging [i]E[/i] moves Line [i]DE[/i] along Ray [i]AB[/i]. Line [i]DE [/i]was constructed to be perpendicular to Ray [i]AB[/i]. This makes Triangle [i]ADE[/i] a right triangle.

As you drag [i]E[/i], Triangle [i]ADE[/i] is changed, but what relationship exists between all these versions of Triangle [i]ADE[/i]?

All of these Triangle [i]ADE[/i]s are right triangles, that's all. All of these Triangle [i]ADE[/i]s are similar, by AA Triangle Similarity. All of these Triangle [i]ADE[/i]s are congruent, by ASA Triangle Congruence.

Note that dragging C changes the measure of Angle A.

As you drag [i]C[/i], Triangle [i]ADE[/i] is changed, but what relationship exists between all these versions of Triangle [i]ADE[/i]?

All of these Triangle [i]ADE[/i]s are right triangles, that's all. All of these Triangle [i]ADE[/i]s are similar, by AA Triangle Similarity. All of these Triangle [i]ADE[/i]s are congruent, by ASA Triangle Congruence.

Exploring Trigonometric Ratios Worksheet (don't worry about this if you've received the worksheet in class)

Use this to complete the tables 1 and 3 on the Exploring Trigonometric Ratios Worksheet

Answer this question after you've completed Table 1.

When you compared the values in Table 1 to values in a trigonometric table, what column did they match with?

sine tangent cosine

Table 1

What did you observe about the multiple trials using the same angle in Table 1?

The ratio got smaller as the triangle got bigger. The ratio got bigger as the triangle got bigger. The ratio stayed the same.

Answer this question after you've completed Table 3.

When you compared the values in Table 3 to values in a trigonometric table, what column did they match with?

tangent sine cosine

Table 3

What did you observe about the multiple trials using the same angle in Table 3?

The ratio got smaller as the triangle got bigger. The ratio stayed the same. The ratio got bigger as the triangle got bigger.

VOCABULARY

Remember that [b]complementary angles[/b] add up to 90 degrees.

Compare the values in table 1 to the values of the complementary angles in table 2.

What do you notice?

The ratios are the same. The ratios have a difference of 0.5 The ratios add up to 1.