Transcript Sections 1.1 and 1.3

Math 115 – Trigonometry
Andy Rosen
www.rosenmath.com
Student Learning Outcomes
• Analytically evaluate the six trigonometric functions of
angles of measures that are multiples of 30 degrees and
45 degrees.
• Be able to solve equations involving trigonometric
functions.
• Use basic identities to verify trigonometric identities or
to simplify trigonometric expressions.
• Use trigonometric functions to solve application
problems involving unknown sides of right triangles.
Angles, Degrees, and Triangles
A ray starts at a point and extends forever in
one direction.
When a ray is rotated around its endpoint, an
angle is created.
initial side
negative
angle
positive
angle
initial side
When graphing in the Cartesian plane, the
initial side is generally the positive x-axis.
One way to measure the size of an angle is by
using degrees. One complete rotation has a
measure of 360°.
• The Greek letter θ (theta) is frequently
used to name an angle
• A right angle measures 90°
• A straight angle measures 180°
• An acute angle measures less than 90°
• An obtuse angle measures between 90°
and 180°
Two angles are complementary is their sum
is 90°. α and β are called complements.
Two angles are supplementary is their sum is
180°. α and β are called supplements.
Ex.
A triangle is a three-sided figure.
Thm. The sum of the angles of a triangle is 180°
Ex. If two angles of a triangle are 16° and 96°,
find the third angle.
An equilateral triangle has three congruent
sides and three equal angles.
An isosceles triangle has two congruent sides
and two congruent angles.
• The congruent sides
B are called legs
LEG
A
LEG
C
A right triangle has one angle that measures
90°.
• The side across from the right angle is
called the hypotenuse.
• The other sides are called legs.
• The most important triangle for this class.
leg
leg
Thm. Pythagorean Theorem
In a right triangle, a2 + b2 = c2, where c is the
length of the hypotenuse, and where a and
b are the lengths of the legs.
Ex. You have a 10-foot ladder and want it to reach a
height of 8 feet when leaned against a wall. How far
from the wall should you put the base of the ladder?
Ex. Find x.
Def. A set of three integers is a Pythagorean
Triple if they satisfy the equation a2 + b2 = c2.
Ex. 3,4,5
6,8,10
5,12,13
8,15,17
50,120,130
32,60,68
Ex. Find x.
x
9
12
50
x
x
24
40
10
This is a good way to save time.
Special Right Triangles
1
45º-45º-90º Triangle
?
30º-60º-90º Triangle
?
2
Ex. A house has a roof with a 45° pitch.
If the house is 60 feet wide, find the
lengths of the sides of the roof that
form the attic. Round to the nearest
foot.
Ex. Find x and y.
8
60
y
x
Trigonometry
Def. Trigonometry is the study of the
relationships between the angles and sides
of a triangle.
Def. A comparison of the lengths of two sides
of a right triangle is called a trigonometric
ratio. The three primary ratios of
trigonometry are sine, cosine, and tangent.
The primary trig ratios can be found by
remembering
SOH
opp
si n  
hyp
CAH
adj
cos 
hyp
TOA
opp
tan  
adj
The other three trig ratios are:
• Cosecant
• Secant
• Cotangent
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Ex. Find the six trig ratios for θ in the triangle
Ex. Find the six trig ratios for θ in the triangle
Ex. Find the requested trig ratios.
5
12

13

sin  
cos  
tan  
cot  
sec  
csc  
Thm. Cofunction Theorem
If α + β = 90°, then
sin α = cos β
tan α = cot β
sec α = csc β
Identities
sin   cos  90   
cos  sin  90   
tan   cot  90   
cot   tan  90   
sec  csc  90   
csc  sec  90   
Ex. sin 30 =
cot 57 =
sec 83 =
5
sin


Ex. In the triangle,
13 and the perimeter
is 60. Find the length of the hypotenuse.