Transcript Angles, Degrees, and Special Triangles

Definition II: Right Triangle
Trigonometry
Trigonometry
MATH 103
S. Rook
Overview
• Section 2.1 in the textbook:
– Right triangle Trigonometry
– Cofunction theorem
– Exact values for common angles
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Right Triangle Trigonometry
Right Triangle Trigonometry
• Another way to view the six trigonometric
functions is by referencing a right triangle
• You must memorize the following definition –
a helpful mnemonic is SOHCAHTOA:
opposite
sine 
hypotenuse
1
hypotenuse
cosecant

sine
opposite
adjacent
cosine
hypotenuse
1
hypotenuse
secant 

cosine
adjacent
opposite
tangent
adjacent
1
adjacent
cotangent

tangent opposite
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Right Triangle Trigonometry
(Continued)
• Definition II is an extension of Definition I as
long as angle A is acute (why?):
– Lay the right triangle on the Cartesian Plane such
that A is the origin and B is the point (x, y)
opp A y
hyp
r
sin A 

csc A 

hyp
r
opp A y
cos A 
adj A x

hyp r
sec A 
hyp r

adj A x
adj A x
opp A y
cot A 

tan A 

opp A y
adj A x
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Right Triangle Trigonometry
(Example)
Ex 1: For each right triangle ABC, find sin A,
csc A, tan A, cos B, sec B, and cot B:
a)
b) C = 90°, a = 3, b = 4
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Cofunction Theorem
Cofunctions
• The six trigonometric functions can be
separated into three groups of two based on
the prefix co:
– sine and cosine
– secant and cosecant
– tangent and cotangent
• Each of the groups are known as cofunctions
• The prefix co means complement or opposite
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Cofunctions and Right Triangles
opp A y adj B
sin A 
 
 cos B
hyp
r
hyp
adj A x opp B
cos A 
 
 sin B
hyp r
hyp
hyp r
hyp
sec A 
 
 csc B
adj A x opp B
hyp
r
hyp
csc A 
 
 sec B
opp A y adj B
opp A y adj B
tan A 
 
 cot B
adj A x opp B
adj A x opp B
cot A 
 
 tan B
opp A y adj B
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Cofunctions and Right Triangles
(Continued)
• The measure of the angles in
a triangle must sum to 180°
• By definition, a right triangle
contains a right angle
measuring 90° (C = 90°)
• Therefore, the remaining two angles must
sum to 90° (A + B = 90°)
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Cofunction Theorem
• Cofunction Theorem: If angles A and B are
complements of each other, then the value of
a trigonometric function using angle A will be
equivalent to its cofunction using angle B or
vice versa
sin A  cos B AND sin B  cos A

If A  B  90 : sec A  csc B AND sec B  csc A
tan A  cot B AND tan B  cot A

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Cofunction Theorem (Example)
Ex 2: Use the Cofunction Theorem to fill in the
blanks so that each equation becomes a true
statement:
a) cot 12° = tan ____
b) sec 39° = csc ____
c) sin 80° = ___ 10°
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Exact Values for Common Angles
Exact Values for Common Angles
• For select angles, we can obtain exact values
for the trigonometric functions:
x
x 3
3
cos30 

2x
2
x
cos 60 
1
2
sin 45 


2
x 2
2
1
2
cos45 


2
x 2
2
x 1

2x 2
sin 30 
x 1

2x 2
sin 60 
x 3
3

2x
2
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Exact Values for Common Angles
(Continued)
• Only need to memorize the sine and cosine
values:
– Can derive the remaining trigonometric functions
through identities
• e.g.
3
sin 60
tan60 
 2  3
1
cos60
2
• Also:
cos 0  1
cos 90  0
sin 0  0
sin 90   1
sin 90 1
tan 90 
  undefined
cos 90 0
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Exact Values for Common Angles
(Continued)
• In summary, this chart MUST be memorized by
chapter 3:
θ
cos θ
sin θ
0°
1
0
30°
3
2
1
2
1
2

2
2
1
2

2
2
60°
1
2
3
2
90°
0
1
45°
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Exact Values for Common Angles
(Example)
Ex 3: For each of the following, replace x with
30°, y with 45°, and z with 60°, and then
simplify as much as possible:
a) 3sin(2y)
b) 2sec(90° – z)
c) 4csc(x)
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Summary
• After studying these slides, you should be able to:
–
–
–
–
Apply the six trigonometric functions to a right triangle
State the definition of a cofunction
Understand and use the Cofunction Theorem
State and use values of the trigonometric functions for
common angles
• Additional Practice
– See the list of suggested problems for 2.1
• Next lesson
– Calculators and Trigonometric Functions of an Acute Angle
(Section 2.2)
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