9.10 Trigonometric Ratios

B

AC

A

BC

Explore1.Find the sine, cosine, and tangent of <A and <B in Δ ABC.

2. Compare sin <A with sin <B Compare cos <A with cos <B Compare tan <A with tan <B

3. Write a generalization that your answers to the problem above suggest.

B

AC

3

4

5sin<A = sin<B =

cos<A= cos<B =

tan<A = tan<B =3

35

45

4

4

3

45

5

3

Conclusion:

The sine of an angle equals the cosine of the complementary angle, or the cosine of an angle equals the sine of the complementary angle.

Also, the tangents of complementary angles are reciprocals.

Angle of ElevationAn angle of elevation is the angle formed from point P to point A.

PA (line of sight) and PH (horizontal line) is called the angle of elevation.

A

PH

A

PH

Angle of depression

If you look down from point P towards point B, the angle that PB makes with PH is called the angle of depression.

HP

BA

ATTENTION!

An angle of elevation or depression is an angle between a LINE OF SIGHT and the HORIZONTAL.

DO NOT USE THE VERTICAL!!!!!

For triangles other than the 30-60-90, and 45-45-90, use the table on page 424.

360m

H

C S

X

What is the distance by land to the ship?

28°

Draw parallel lines from the ground. Remember alt int <‘s are congruent.

<CSH = 28°

Thus: tan 28 ° = 360

0.5317 = 360

0.5317X = 360

X ≈ 677

xLook up tan 28 ° on the table, page 424, and plug it in to solve.

x

Multiply both sides by x

Divide by 0.5317

The horizontal distance is about 677 m.

Use a calculator where necessary.