trig functions of acute angles
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TRIG FUNCTIONS OF ACUTE ANGLESSection 12-2
Pages 555-560
Trig Functions of Acute Angles
• Draw an acute angle θ in standard position.• Choose any point (x, y) on the terminal side of θ and let r be the distance from the origin to (x, y)
θ
(x, y)
x
yr
Trig Functions of Acute Angles
• With this triangle, we have the following definitions:
Sin θ = Cos θ = Tan θ =
θ
(x, y)
x
yr
Trig Functions of Acute Angles• Another way to think of this is using the word:
SohCahToa
θAdjacent
OppositeHypotenuse Sin θ =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Trig Functions of Acute Angles• Another way to think of this is using the word:
SohCahToa
θAdjacent
OppositeHypotenuse Cos θ =
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Trig Functions of Acute Angles• Another way to think of this is using the word:
SohCahToa
θAdjacent
OppositeHypotenuse Tan θ =
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Trig Functions of Acute Angles
• Find the Sin θ, Cos θ, and Tan θ given that θ is an angle in standard position passing through the point (4, 5).
4
5√𝟒𝟏θ
Sin θ =5
√415√4141=
Cos θ =4
√414 √4141=
Tan θ =54
Trig Functions of Acute Angles
• In addition to the Sin θ, Cos θ, and Tan θ, there are three reciprocal functions
1) Cosecant θ (Csc θ) =
2) Secant θ (Sec θ) =
3) Cotangent θ (Cot θ) =
Trig Functions of Acute Angles
• Find the value of the six trig functions of θ whose terminal side passes through (5, 12)
(𝟓 ,𝟏𝟐)
5
1213
θ
Sin θ =1213
Cos θ =513
Tan θ =125
Csc θ =1312
Sec θ =135
Cot θ =512
Trig Functions of Acute Angles
• Find the value of the six trig functions of θ whose terminal side passes through (2, 4)
(𝟐 ,𝟒)
2
42
θ
Sin θ =2√55
Cos θ =√55
Tan θ = 2
Csc θ =√52
Sec θ =√5
Cot θ =12
Trig Functions of Acute Angles• Oral Exercises 1-4• Page 559, 1-8
TRIG FUNCTIONS OF ACUTE ANGLESSection 12-2
Pages 555-560
Trig Functions of Acute Angles
• Find the value of the six trig functions of θ whose terminal side passes through (-3,- 4)
(−𝟑 ,−𝟒)
-4
-3
5
θ Sin θ =−35
Cos θ =−45
Tan θ =43
Csc θ =−53
Sec θ =−54
Cot θ =34
Trig Functions of Acute Angles
• A trigonometric equation that is true for all values of θ is called a trigonometric identity
• Quotient Identities
Tan θ = Cot θ =
Trig Functions of Acute Angles
• Pythagorean Identities
a) Sin2θ + Cos2θ = 1
b) 1 + Tan2θ = Sec2θ
c) 1 + Cot2θ = Csc2θ
Trig Functions of Acute Angles
• Using the identities, find the value of the remaining trig functions if Sin θ =
Sin2θ + Cos2θ = 12 + Cos2θ = 1
+ Cos2θ = 1
Cos2θ =
Cos θ =
Csc θ = 3
Sec θ = =
Trig Functions of Acute Angles
• Using the identities, find the value of the remaining trig functions if Sin θ =
Cos θ =
Cot θ = 2
Sin θ =
Tan θ =
Csc θ = 3
Sec θ =
Trig Functions of Acute Angles
• Using the identities, find the value of the remaining trig functions if Cos θ =
Sin2θ + Cos2θ = 12θ + 2 = 1
Sin2θ + = 1
Sin2θ =
Sin θ =
Csc θ =
Sec θ =
Trig Functions of Acute Angles
• Using the identities, find the value of the remaining trig functions if Sin θ =
Cos θ =
Cot θ =
Sin θ =
Tan θ =
Csc θ =
Sec θ =
Trig Functions of Acute Angles
• Cofunction Identities• The Sine and Cosine functions are called cofunctions.
A
B
C
a
b
c
Sin A = 𝑎𝑐
Cos B = 𝑎𝑐
= Cos B
What can we say about angles A and B?→ Complementary
Trig Functions of Acute Angles
• Cofunction Identities
A
B
C
a
b
c
Sin A = 𝑎𝑐 = Cos B
Tan A = 𝑎𝑏 = Cot B
Csc A = 𝑐𝑎 = Sec B
Trig Functions of Acute Angles
• Cofunction Identities
From these identities, note that:
Sin θ = Cos (90° – θ)
Tan θ = Cot (90° – θ)
Sec θ = Csc (90° – θ)
Cos θ = Sin (90° – θ)
Cot θ = Tan (90° – θ)
Csc θ = Sec (90° – θ)
Trig Functions of Acute Angles• Oral Exercises 5-8• Page 559, 9-19
TRIG FUNCTIONS OF ACUTE ANGLESSection 12-2
Pages 555-560
Trig Functions of Acute Angles
• So far in this section:
a) Soh Cah Toa
b) Identities
a) Quotient Identities
b) Pythagorean Identities
c) Cofunction Identities
Trig Functions of Acute Angles
• In trigonometry, there are two triangles that are used repeatedly in different problems.
1) 30-60-90 right triangle
2) 45-45-90 right triangle
30°
60°12
√3
45°
45°
1
1
√2
Trig Functions of Acute Angles
• Using these two triangles, we can find the following:
Sin θ
Cos θ
Tan θ
30° 45° 60°12
12
√22
√22
√32
√32
√33 √31
Trig Functions of Acute Angles
• Use the two triangles (or chart) to evaluate the reciprocal functions.
Csc θ
Sec θ
Cot θ
30° 45° 60°
2
2√2
√2 2√33
2√33
√3√331
Trig Functions of Acute Angles
• Find the lengths of the missing sides.
30 °12
𝑏𝑐
𝑇𝑎𝑛30 °=¿ √33
𝑏12
=¿
3𝑏=12√3𝑏=4√3
Cos√32
12𝑐
=¿
24
𝑐=24
√3¿ 24√3
3 ¿8 √3
Trig Functions of Acute Angles
• Find the lengths of the missing sides.
6 0°
5
𝑏𝑐
𝑇𝑎𝑛60 °=¿ √35𝑏
=¿
5
𝑏=5√33
Sin√32
5𝑐=¿
10
𝑐=10
√3¿ 10√3
3
Trig Functions of Acute Angles
26) Find the length of x.
x45° 60°
15
Sin√22
h15
=¿
2 15
h=15√22
h
x' x'¿15√22
Trig Functions of Acute Angles
26) Find the length of x.
x45° 60°
15
Tan √315√22
x ''=¿
15√22x ′′
=√3
x ′′=15 √22√3
15√22
x'' =15 √66
=5 √62
Trig Functions of Acute Angles
26) Find the length of x.
45° 60°
15
15√22
5 √62
X = 15√22 +
5 √62
=
Trig Functions of Acute Angles
28) Find the length of x.
30°
15°
24
x
x'
Tan 1x '24
=¿
24
Trig Functions of Acute Angles
28) Find the length of x.
30°
15°
24
x
24 -
Tan√33
x '′24
=¿
24
x
8
X =
24
8
Trig Functions of Acute Angles• Oral Exercises 9-12• Page 559, 20-34
Trig Functions of Acute Angles
Trig Functions of Acute Angles
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