TRIG FUNCTIONS OF ACUTE ANGLES Section 12-2 Pages 555-560

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


• 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


SohCahToa

θAdjacent

OppositeHypotenuse Cos θ =

𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒


SohCahToa

θAdjacent

OppositeHypotenuse Tan θ =

𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡


• 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


• 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) =


• 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


• 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


Pages 555-560


• 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


• A trigonometric equation that is true for all values of θ is called a trigonometric identity

• Quotient Identities

Tan θ = Cot θ =


• Pythagorean Identities

a) Sin2θ + Cos2θ = 1

b) 1 + Tan2θ = Sec2θ

c) 1 + Cot2θ = Csc2θ


• 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 = =



Cos θ =

Cot θ = 2

Sin θ =

Tan θ =

Csc θ = 3

Sec θ =


• 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 =



Cos θ =

Cot θ =

Sin θ =

Tan θ =

Csc θ =

Sec θ =


• 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


• Cofunction Identities

A

B

C

a

b

c

Sin A = 𝑎𝑐 = Cos B

Tan A = 𝑎𝑏 = Cot B

Csc A = 𝑐𝑎 = Sec B


• 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)



Pages 555-560


• So far in this section:

a) Soh Cah Toa

b) Identities

a) Quotient Identities

b) Pythagorean Identities

c) Cofunction Identities


• 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


• 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


• 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


• 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


• 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


26) Find the length of x.

x45° 60°

15

Sin√22

h15

=¿

2 15

h=15√22

h

x' x'¿15√22



x45° 60°

15

Tan √315√22

x ''=¿

15√22x ′′

=√3

x ′′=15 √22√3

15√22

x'' =15 √66

=5 √62



45° 60°

15

15√22

5 √62

X = 15√22 +

5 √62

=



30°

15°

24

x

x'

Tan 1x '24

=¿

24



30°

15°

24

x

24 -

Tan√33

x '′24

=¿

24

x

8

X =

24

8




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TRIG FUNCTIONS OF ACUTE ANGLES Section 12-2 Pages 555-560