Right Triangle Trigonometry Definitions

(Instructor Notes)

This activity is designed for a 50 min. class.

Materials:

Triangles – Print out the last 10 pages of this document. It helps to use different

colors as indicated. All Blue triangles have a leg that should measure 6 cm.

Red, 12 cm. Green, 18 cm. Yellow, 24 cm.

Rulers with cm markings. It helps to have 1 per person, but 2 rulers per group of 4

can also work if you are short.

Activity (pages 2-4 of this document)

Before Class:

Make sure tables are in groups of 4 before class begins.

Have 1 color triangle to be picked as the first students come in. (or have 1 color already at

each of the group tables)

Other students coming in can pick from 2 colors. Save 1 color for the end – last students will

pick from the last color. This is so there aren’t any groups with just 1 person.

Students find their seat based on the table that has a triangle with angles that match theirs.

Give a brief introduction to the definitions of trig ratios and the Pythagorean theorem.

Make sure students know to measure the legs of their triangles, but NOT the hypotenuse

when sketching their triangle on the worksheet.

Then have students complete the worksheet with their group. They can start on their own,

but will need to compare answers to their group for a couple of questions.

Important note about printing and measuring:

Print shapes on different color paper. Each group should have up to 4 different color

triangles. Make the largest triangle in each group the same color (maybe yellow), the next

largest the same color (maybe green), etc. Since cutting out or printing the triangle shapes

can lead to some triangles being slightly off on a side, be sure to give students instructions

about rounding. The triangles included should all have legs that are a whole or half cm, so

have the students measure in cm rather than inches.

Inform students: “All legs of your triangles should be rounded to the nearest whole or half

cm. Do NOT measure the hypotenuse: Follow instructions given on the worksheet!”

Right Triangle Trigonometry Definitions

Objectives:

I can find the six trig ratios of an angle from a right triangle.

I can solve for any side of a right triangle given an angle and another side.

Key Terms and Formulas:

The following formulas apply ONLY to RIGHT triangles. You can NOT use them for other

types of triangles.

sin 𝜃 =𝑜𝑝𝑝

ℎ𝑦𝑝 cos 𝜃 =

𝑎𝑑𝑗

ℎ𝑦𝑝 tan 𝜃 =

𝑜𝑝𝑝

𝑎𝑑𝑗

csc 𝜃 =ℎ𝑦𝑝

𝑜𝑝𝑝 sec 𝜃 =

ℎ𝑦𝑝

𝑎𝑑𝑗 cot 𝜃 =

𝑎𝑑𝑗

𝑜𝑝𝑝

Pythagorean Theorem: 𝑎2 + 𝑏2 = 𝑐2

1. Make a sketch of your triangle here.

2. Measure the LEGS of your actual triangle and label them in the figure. (Do NOT measure the

hypotenuse) Note: Your instructor may have special instructions about rounding.

3. Use the Pythagorean theorem to find the hypotenuse and label it with its exact value.

4. Label the smallest angle A, the middle angle B, and the 90O angle C.

5. Find the following based on your triangle. Use EXACT values and REDUCE your answers.

Compare your answers with the others in your group.

sin 𝐴 = csc 𝐴 =

cos 𝐴 = sec 𝐴 =

tan 𝐴 = cot 𝐴 =

What do you notice about your answers and the answers of others in your group?

6. Suppose another group told you that for their triangles, 2

sin5

A .

a) Is it possible that none of the triangles has a leg length 2 and hypotenuse with length 5? b) Sketch this triangle and label the leg with 2 and the hypotenuse 5.

c) Use Pythagorean Theorem to find the exact value of the other leg – label it on the sketch.

d) Use your sketch to find the exact value of the other trig functions of A.

sin 𝐴 = csc 𝐴 =

cos 𝐴 = sec 𝐴 =

tan 𝐴 = cot 𝐴 =

e) Is it possible that a triangle could have different lengths on its edges but the same trig

ratios? If so, how could this happen? If not, why not?

7. Find the missing lengths on the following triangles. Round to 2 decimal places.

5 cm

40O

x

8 cm 20O

x

12.5 cm

75O

x

15 cm 60O

x

2.3 cm 75O

x 4 cm

30O

x

Blue

Red

Red

Green

Green

Green

Yellow

Yellow

Yellow

Yellow