Trigonometry and Vectors

1. Trigonometry, triangle measure, from Greek.2. Mathematics that deals with the sides and angles of triangles,

and their relationships.3. Computational Geometry (Geometry – earth measure).4. Deals mostly with right triangles.5. Historically developed for astronomy and geography.6. Not the work of any one person or nation – spans 1000s yrs.7. REQUIRED for the study of Calculus.8. Currently used mainly in physics, engineering, and chemistry,

with applications in natural and social sciences.

Background – Trigonometry

Trigonometry and Vectors

1. Total degrees in a triangle:2. Three angles of the triangle below:3. Three sides of the triangle below:4. Pythagorean Theorem:

x2 + y2 = r2

a2 + b2 = c2

Trigonometry

180

A

B

C

r, y, and x

y

x

r

HYPOTENUSE

A, B, and C

Trigonometry and Vectors

State the Pythagorean Theorem in words:“The sum of the squares of the two sides of a right triangle is

equal to the square of the hypotenuse.” Pythagorean Theorem:

x2 + y2 = r2

Trigonometry

A

B

C

y

x

r

HYPOTENUSE

Trigonometry and Vectors

NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

1. Solve for the unknown hypotenuse of the following triangles:

Trigonometry – Pyth. Thm. Problems

4

3?a)

1

1?b)

1?c)

3222 ba c

22 bac 169

5c

22 bac 22 11

2c

22 bac 22 1)3(

2c 13

Align equal signs when possible

Trigonometry and Vectors

Common triangles in Geometry and Trigonometry

3

4

5

1

Trigonometry and VectorsCommon triangles in Geometry and

Trigonometry

11

1

2

45o

45o

2

3

30o

60o

You must memorize these triangles

2 3

Trigonometry and Vectors

NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

2. Solve for the unknown side of the following triangles:

Trigonometry – Pyth. Thm. Problems

8

?

10 ?

15

?

12

13 12a) b) c)

22 bca

36 6a

222 ba c 222 bc a

22 801

22 bca 22 2113

144169 25

5a

22 bca 22 2115

144225 81

9a

Divide all sides by 2 3-4-5 triangle

Divide all sides by 3 3-4-5 triangle

Trigonometry and Vectors

1. Standard triangle labeling.2. Sine of <A is equal to the side opposite <A divided by the

hypotenuse.

Trigonometric Functions – Sine

A

B

C

y

x

r

HYPOTENUSE

OPP

OSI

TEADJACENT

sin A = yr

sin A = opposite

hypotenuse

Trigonometry and Vectors

1. Standard triangle labeling.2. Cosine of <A is equal to the side adjacent <A divided by the

hypotenuse.

Trigonometric Functions – Cosine

A

B

C

y

x

r

HYPOTENUSE

OPP

OSI

TEADJACENT

cos A = xr

cos A = adjacent

hypotenuse

Trigonometry and Vectors

1. Standard triangle labeling.2. Tangent of <A is equal to the side opposite <A divided by the

side adjacent <A.

Trigonometric Functions – Tangent

A

B

C

y

x

r

HYPOTENUSE

OPP

OSI

TEADJACENT

tan A = yx

tan A = opposite adjacent

Trigonometry and Vectors

3

4

51

2

3

1

1

2

NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

3. For <A below calculate Sine, Cosine, and Tangent:

Trigonometric Function Problems

A

B

C A

B

CA

B

C

a) b) c)

sin A = opp. hyp. cos A = adj.

hyp.tan A =

opp. adj.

Sketch and answer in your notebook

Trigonometry and Vectors

3

4

5

3. For <A below, calculate Sine, Cosine, and Tangent:

Trigonometric Function Problems

A

B

C

a) sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

tan A = opposite adjacentsin A = 3

5

cos A = 45

tan A = 34

Trigonometry and Vectors

3. For <A below, calculate Sine, Cosine, and Tangent:

Trigonometric Function Problems

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

tan A = opposite adjacentsin A = 1

√2

cos A =

tan A = 1

1

1

2

A

B

C

b)

1 √2

Trigonometry and Vectors

3. For <A below, calculate Sine, Cosine, and Tangent:

Trigonometric Function Problems

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

tan A = opposite adjacentsin A = 1

2

cos A =

tan A =

√3 2

12

3A

B

C

c)

1 √3

Trigonometry and Vectors

Trigonometric functions are ratios of the lengths of the segments that make up angles.

Trigonometric Functions

tan A = opposite adjacent

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

Trigonometry and Vectors

Common triangles in Trigonometry

1

1

2

45o

45o

12

3

30o

60o

You must memorize these triangles

Trigonometry and Vectors

Trigonometric FunctionsNO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

4. Calculate sine, cosine, and tangent for the following angles:a. 30o

b. 60o

c. 45o

12

3

30o

60osin 30 =

12

cos 30 = √3 2

tan 30 = 1 √3

Trigonometry and Vectors

Trigonometric FunctionsNO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

4. Calculate sine, cosine, and tangent for the following angles:a. 30o

b. 60o

c. 45o

12

3

30o

60o

cos 60 = 12

sin 60 = √3 2

tan 60 = √3

Trigonometry and Vectors

Trigonometric FunctionsNO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

4. Calculate sine, cosine, and tangent for the following angles:a. 30o

b. 60o

c. 45o

tan 45 = 1

sin 45 = 1 √2

cos 45 = 1 √2

1

1

2

45o

45o

Unless otherwise specified:

• Positive angles measured counter-clockwise from the horizontal.

• Negative angles measured clockwise from the horizontal.

• We call the horizontal line 0o, or the initial side

0

90

180

270

Trigonometry and VectorsMeasuring Angles

30 degrees

45 degrees

90 degrees

180 degrees

270 degrees

360 degrees

INITIAL SIDE

-330 degrees

-315 degrees

-270 degrees

-180 degrees

-90 degrees

=

=

=

=

=

Trigonometry and Vectors

Begin all lines as light construction lines!• Draw the initial side – horizontal line.• From each vertex, precisely measure the angle with a protractor.• Measure 1” along the hypotenuse. Using protractor, draw vertical

line from the 1” point.• Darken the triangle.

Trigonometry and Vectors

CLASSWORK / HOMEWORK

Complete problems 1-3 on the Trigonometry Worksheet