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