2.1: Cofunctions, 45-45-90, & 30-60-90 Triangles
1. The Trigonometric Cofunctions
2. Right-Triangle-based definitions of the trig functions
3. The Cofunction Identities
4. Examples using the Cofunction Identities
5. The values of sin, cos, and tan for acute angles
6. 45-45-90 and 30-60-90 triangles
HW: Pg. 68 #1-7 all; 11-39 odd (skip 15)
MK: 2.1 Makeup Homework from website
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1. Trigonometric Cofunctions
Recall:
r
!
P(x, y)
sin ! = y / r
csc ! = r / y, y " 0
y
cos ! = x / r
sec ! = r / x, x " 0
tan ! = y / x, x " 0
cot ! = x / y, y " 0
x
def: The trigonometric cofunctions are the pairs sine/cosine, secant/cosecant,
and tangent/cotangent.
The Trig Cofunctions:
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sin ! = y / r
cos ! = x / r
sec ! = r / x, x " 0
csc ! = r / y, y " 0
tan ! = y / x, x " 0
cot ! = x / y, y " 0
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2. Right-Triangle-based definitions of the trig functions
Recall: cofunctions
r
!
x
P(x, y)
sin ! = y / r
cos ! = x / r
y
sec ! = r / x, x " 0
csc ! = r / y, y " 0
tan ! = y / x, x " 0
cot ! = x / y, y " 0
Recall: def: complementary angles are a pair of angles whose measures add
to 90°.
B
(r) c
A
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Notice: !A & !B are complementary
a (y)
b
(x)
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C
sin A = opp / hyp = a/c
cos B = adj / hyp = a/c
sec A = hyp / adj = c/b
csc B = hyp / opp = c/b
tan A = opp / adj = a/b
cot B = adj / opp = a/b
Conclusion: Cofunctions values of complementary angles are equal.
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3. The Cofunction Identities
B
Recall:
c
A
a
b
C
sin A = opp / hyp = a / c
cos B = adj / hyp = a / c
sec A = hyp / adj = c / b
csc B = hyp / opp = c / b
tan A = opp / adj = a / b
cot B = adj / opp = a / b
Note: m!B = 90! " m!A
The Cofunction Identities: sin A = cos(90! ! A)
cos A = sin(90! ! A)
sec A = csc(90! ! A)
csc A = sec(90! ! A)
tan A = cot(90! ! A)
cot A = tan(90! ! A)
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4. Examples using the Cofunction Identities
Example 1: Write the each of the following in terms of its cofunction.
(a) sin 88°= cos(90°-88°)=cos 2°
(b) sec 18°= csc(90°-18°)=csc 72°
(c) tan 25°= cot(90°-25°)=cot 65°
Example 2: Find the value of x such that cos(x + 4°) = sin(3x+2°),
assuming both angles are acute.
The cofunction values will be equal if the angles are complementary.
(x + 4) + (3x+2) = 90
4x + 6 = 90
x = 21°
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5. A The values of sin, cos, and tan for acute angles
Recall:
r
!
P(x, y)
sin ! = y / r
y
cos ! = x / r
tan ! = y / x, x " 0
x
As ! increases from 0° to 90°, y increases, x decreases, & r is constant and …
sin ! increases from 0 to 1
cos ! decreases from 1 to 0
tan ! increases from 0 to +!
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6. 45-45-90 & 30-60-90 triangles
B
Recall:
sin A = opp / hyp = a / c
c
A
1
a
C
b
2
45°
cos A = adj / hyp = b / c
tan A = opp / adj = a / b
2
30°
3
45°
1
60°
1
1
2
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2
!
sin !
cos !
30°
1/ 2
3/2
45°
2 /2
2 /2
60°
3/2
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2.1 Summary
B
c
A
The Cofunction Identities:
a
C
b
sin A = cos(90! ! A)
cos A = sin(90! ! A)
sec A = csc(90! ! A)
csc A = sec(90! ! A)
tan A = cot(90! ! A)
cot A = tan(90! ! A)
As ! increases from 0° to 90° sin ! increases from 0 to 1
cos ! decreases from 1 to 0
tan ! increases from 0 to +!
1
2
45°
2
3
60°
45°
1
2
30°
1
1
2
!
sin !
cos !
30°
1/ 2
3/2
45°
2 /2
2 /2
60°
3/2
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