right triangle trigonometry:

Right TriangleTrigonometry:

Word SplashUse your prior knowledge or make up a meaning for the following words to create a story. Use your imagination!

hypotenus

etrigonometric ratios

cosine

sinetangent

similar

scale factor

congruent

correspondi

ng

surveyingastronomy

angle of elevation

angle of

depression

direct

measurement

Relating to the Real World

• Before any spacecraft ever traveled to another planet, astronomers had figured out the distance from each planet to the sun. They accomplished this feat by using trigonometry the mathematics of triangle measurement. You will learn how to use trigonometry to measure distances that you could never otherwise measure.

Measure for Measure

• What do the Trigonometric Ratios tell you about the parts of a triangle?

• What are the conditions for triangles to be similar?

• How can you remember the Trigonometric ratios?

Label These Two Triangles

Angle A

Angle A

The Tangent Ratio

• The word Trigonometry comes from the Greek words meaning “triangle measurement.” A ratio of the lengths of sides of a right triangle. This ratio is called the TANGENT.

• TANGENT OF A = leg opposite A leg adjacent to A

• Slope of a line = Rise Run

Using the Tangent Ratio

AC

B

Leg Adjacent to < A, CA

Leg Opposite to < A, BC

Example

Tan U = opposite = TV = 3 adjacent UV 4

T

V U

35

4

Write the Tangent Ratios for < U and < T.

Tan T = opposite = UV = 4 adjacent TV 3

Try This

• Write the Tangent ratios for < K and < J

• How is Tan K related to the Tan J ?J

L

K

3

7

Try This: Find the Tangent of < A to the nearest tenth

1. 2.

Hint: Find the Ratio First!

4

5

8

A

A

4

Practice:• Find the Tan A and Tan B ratios of each triangle.

A

B

3 5

4AB

25

4

AB

7

B

A

6

10

5

Using the Sine Ratio

AC

B

Leg Opposite to < B, CA

Leg Opposite to < A, BC

Hypotenuse, AB

Example

Sin U = opposite = TV = 3 hypotenuse TU 5

T

V U

35

4

Write the Sine Ratios for < U and < T.

Sin T = opposite = UV = 4 hypotenuse TU 5

Try This

• Write the Sine ratios for < K and < J

J

L

K

6

8

10

Practice:• Find the Sin A and Sin B ratios of each triangle.

A

B

3 5

4AB

24

5

AB

7

B

A

8

10

7

Using the Cosine Ratio

AC

B

Leg Adjacent to < A, CA

Leg Adjacent to < B, BC

Hypotenuse, AB

Example

Cos U = adjacent = UV = 4 hypotenuse TU 5

T

V U

35

4

Write the Cosine Ratios for < U and < T.

Cos T = adjacent = TV = 3 hypotenuse TU 5

Try This

• Write the Cosine ratios for < K and < J

J

L

K

6

8

10

Practice:• Find the Cos A and Cos B ratios of each triangle.

A

B

3 5

4AB

24

5

AB

7

B

A

6

10

5

Finding Angles

AC

B

Leg Adjacent to < A, CA

Leg Opposite to < A, BC

Hypotenuse, AB

Practice:• Use the Cos-1 and Sin -1 ratios to find the angles of each

triangle.

A

B

3 6

4AB

25

AB

7

B

A

6

10

7