Download Ch 1 Notes - El Camino College

Math 170
Trigonometry Lecture Notes
Chapter 1
1.1 & 1.2 Angles, Degrees, Special Triangles, Distance Formula:
Types of Angles:
Acute
Right
Straight
Obtuse
Complementary Angles: Two angles that add to ____________
Supplementary Angles: Two angles that add to ____________
Find the supplement & complement (if they exist):
a) 73⁰
b) 31⁰
c) 111⁰
d) x⁰
Sum of the Angles in a Triangle is: ___________
Use your knowledge of angles to solve for x and find the angle measures:
1)
xº
2)
(3x + 6)º (2x)º
(1.5x + 10)º
xº
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Math 170
Trigonometry Lecture Notes
Chapter 1
Angles in Trigonometry:
• All angles start at the positive x – axis.
o This is called: Standard Position
o The initial side:
o The terminal side:
• Positive angles are counterclockwise.
• Negative angles are clockwise.
• Full circle is _____o. Straight line is ____o. Right angle is ____o .
Coterminal Angles:
• Any angle in standard position that ends at the same ray.
• Differ by a multiple of __________. For any given ray, there are an infinite number of
coterminal angles.
a) Find 5 angles coterminal with 82⁰.
b) Find 5 angles coterminal with -220⁰
Geometry Review
Vertical Angles:
(5x - 129)°
(2x - 21)°
Types of Triangles:
based on Angles:
Acute
Right
Obtuse
based on Sides:
Equilateral
Isosceles
Scalene
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Math 170
Trigonometry Lecture Notes
Chapter 1
Recall Similar Triangles:
* Corresponding angles are ____________.
* Corresponding sides are ______________.
Given the following similar triangles, find the length of the missing sides:
a)
b)
10’
14’
a
b
a
30"
20’
60’
50"
100"
Use similar triangles to solve the following. Draw the picture first!
a) A tree casts a shadow 30’ long. At the same time a 6’ tall man casts a shadow 4’ long. Find
the height of the tree.
b) A tower casts a shadow 150’ long. At the same time a 5’ tall woman casts a shadow 7.5’
long. Find the height of the tower.
Pythagorean Theorem:
Used to find third unknown side of right triangle
Common Pythagorean Triples:
3:
5:
8:
7:
9:
4
12
15
24
40
:5
:13
:17
:25
:41
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Math 170
Trigonometry Lecture Notes
Practice
a) Given one leg is 5cm and the
hypotenuse is 20 cm
Chapter 1
b) Given legs are 10 in and 24 in
Distance Formula
Really just a fancy form of _________________________
Practice:
1) Find the distance between (2, -3) and ( -1, -2)
Special Right Triangles:
30o – 60o – 90o
Isosceles Right Triangle
Example: If the long leg of 30-60-90 triangle is 8cm, find the length of the other sides.
Rational Expression Review:
1)
Simplify
252 =
2) Rationalize the denominator
12
10
CA 1.2 pg 25 #69, 73; 1.1 pg 12-13 #27, 41, 53, 67;
3) Solve for x
3 x=7
Special Δ worksheet
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Math 170
Trigonometry Lecture Notes
Chapter 1
Similar Triangles are the basis for trigonometric functions. We know that if we have similar
triangles we always have the same ratio of the sides. These ratios are actually given by
trigonometric functions so you won’t have to have the “other triangle” to compare to: the trig
function gives you the ratio of the similar triangle. However, to use them we need (1) a right
triangle and (2) the measure of one more angle in the triangle
1.3 Trigonometric Functions & 1.2 Coordinate System
SOH CAH TOA
Right Triangles
Angles (in Standard Position)
y
P (x, y)
θ
x
Function
Right
Triangles
Angles
Function
sin θ
csc θ
cos θ
sec θ
tan θ
cot θ
Right
Triangles
Angles
Find the 6 trigonometric functions of ϴ, for the following points on the terminal side of ϴ:
• Draw the triangle.
• Find r (use Pythagorean Thm)
a) P (-3, 4)
b) P (-5, - 12)
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Math 170
Trigonometry Lecture Notes
Chapter 1
Signs in each Quadrant: All Students Take Calculus
Function Values of Quadrantal Angles:
θ
0⁰/ 360⁰
90⁰
180⁰
270⁰
sin θ
csc θ
cos θ
sec θ
tan θ
cot θ
Find the following: (No calculator)
a) 4 csc 270⁰ + 3 cos 180⁰
b) sin 180⁰ + cos2 180⁰
Slopes, lines, parabolas
Review on your own:
pg 15-16
Graphing lines: y = mx + b
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Math 170
Trigonometry Lecture Notes
Chapter 1
Using Trig Functions
Find the remaining trig functions, given one trig fn and the quadrant.
• Sketch an angle in the appropriate quadrant.
• Draw a triangle and use the given trig fn to find two sides of the triangle.
• Use Pythagorean Thm. to find the other side.
a) cos θ =
5
θ is in Quad. IV
8
b) csc θ = 2 θ is in Quad. II
c) sin θ =
2
6
d) tan θ = −⅔ sin θ > 0
cos θ < 0
Range of Trigonometric Fns: (pg 29)
• Sin and Cos are between ________ and _________
• Csc and Sec are less than ________ or greater than ___________
• Tan and Cot can be any value.
Is it possible?
a) sin θ = −¾
b) cos θ = 2.5
c) tan θ = 150
d) csc θ = −0.5
CA: 1.3 pg 32 #45, 59, 65, 69; 1.2 pg 25 #21, 48
1.4: 15, 19, 23, 33, 41
1.5: 5, 9, 19, 31, 37, 51, 59, 67, 75, 93
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Math 170
Trigonometry Lecture Notes
Chapter 1
HW Page: 1.4 Introduction to Trig Identities
Find these identities in your book and write them here.
Reciprocal Identities:
(pg 34)
Ratio Identities:
csc ϴ =
sin ϴ =
tan θ
sec ϴ =
cos ϴ =
cot θ
tan ϴ =
cot ϴ =
(pg 35)
Pythagorean Identities
Watch videos from course webpage:
http://www.elcamino.edu/faculty/ammartinez/Math170.html
• Trig Identities A L2 (4:49)
• Trig Identities B L3 (4:57)
Take notes on the proofs and then write the identities in the box.
(Do NOT just write the identities.)
Notes:
The Three Pythagorean Identities
CA: Do the following using identities (NOT Pythagorean Theorem):
1.4 15, 19, 23, 33, 41
1.5 5, 9, 19, 31, 37, 51, 59, 67, 75, 93
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