EPANDING THE CONTENT OF AN OUTLINE using notes.pptx
Chap5 sec5.2
1. Section: 5.2Trigonometric Functions of AtAlAcute AnglesObjectives of this SectionFidthVlfTitiFtifAtAl•Find the Value of Trigonometric Functions of Acute Angles•Use the Fundamental Identities•Find the Remaining Trigonometric Functions Given the Value of One•Use the Complementary Angle Theorem
2. A triangle in which one angle is a right lilldihttilThidangle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legsof the triangle.ggcb 90oa
4. The six ratios of the sides of a right triangle are calledtrigonometricfunctionsofacutecalled trigonometric functions of acute anglesand are defined as follows:
sine of
θ
Name
sin θ
Abbreviation Value
b / c
θ
θ
f
cosine of
θ
θ
tan
cos
b
a c
/
/
θ
cosecant of
tangent of
θ
csc
c b
a
/
θ
θ
cotangent of
secant of
θ
θ
cot
sec
a b
c a
/
/
5. The Six Trigonometric Functions
= = b = = a
H
cos Adjacent
H
sinθ Opposite θ
c c
Hypotenuse Hypotenuse
b
c
a
= = b = =
Opposite
csc Hypotenuse
Adjacent
tanθ Opposite θ
b
a
a
= = c = =
Opposite
cot Adjacent
Adjacent
secθ Hypotenuse θ
6. Find the value of each of the six trigonometricfnctionsoftheangleθtrigonometric functions of the angle .
c = Hypotenuse = 13
b = Opposite = 12
12 13
pp
a2 +b2 =c2
2 2 2
θ
a2 +122= 132
2 169 144 25
Adjacent
a2 =−= a = 5
9. b2 +a2 =c2
c
2
2
2
2
2
b2 a c b + =
90o
c2 c2 c2
2 2 ⎜ ⎛
⎜ ⎛
b a
a
= 1
⎠
⎞
⎜⎝
+ ⎛
⎠
⎞
⎜⎝
⎛
c c
tan 2θ + 1 = sec 2θ
1 + cot 2 θ = csc 2 θ
sin2θ + cos2θ =1
10. Given that cosθ = 1 and θ
is an acute angle
find the exact valu e of each of the remaining
angle,
4
five trigonome tric function of θ .
sin 2θ + cos 2θ = 1
2
4
sin 1 1 ⎟
⎠
⎞
⎜⎝
θ = − ⎛
1
4
sin 1
2
2 ⎟ =
⎠
⎞
⎜
⎝
θ + ⎛
2
2
sin 1 1 ⎟
⎞
⎜
θ = − ⎛ 4
15
16
sin θ = 1 − 1 =
⎝
4
⎠
11. θ 1 i θ 154
; sin 15 4
cos= =
15
4 15
15
4
4
15
1
sin
csc = 1 = = =
θ
4 θ
4
1
1
cos
sec = 1 = =
θ
θ
15
1
4
4
15
1
4
sin 15 tan = = = ⋅ =
θ
θ
θ 15
15
15
1
tan
cot = 1 = =
θ
θ
4 cos
12. Complementary Angles TheoremCofunctions of complementary angles are equal.
θ ( Degrees)
sinθ = cos(90o −θ )
g )
( ( )
θ ( θ
)
cosθ = sin 90o −θ
θ ( θ )
= −
= −
o
o
cot tan 90
tan cot 90
( )
θ ( θ )
θ = −θ
o
o
90
sec csc 90
csc= sec(90o −
13. θθ (Radians)
sinθ = cos(π 2 −θ )
( )
θ ( θ
)
θ = π −θ
t t 2
cos sin 2
θ ( π θ )
π = −
= −
cot tan 2
tan cot ( )
θ ( θ )
θ = π −θ
2
sec csc 2
csc= sec(π −
14. sin 56 o = cos(90o − 56o ) = cos 34 o
sec
π ⎜ ⎛
π π 3π
5
⎠
⎞
⎜⎝
= ⎛ −
2 5
csc
10
= csc
15. Section 5.3
Computing the Values of
TrigFunctionsofAcute Trig Functions of Acute
Angles g
Objectives of this Section•Find the Exact Value of the Trigonometric Functions of 45 DegreesFidhEVlfhTiiFif30•Find the Exact Value of the Trigonometric Functions of 30 Degrees and 60 Degrees•UseaCalculatortoApproximatetheValueofthe•Use a Calculator to Approximate the Value of the Trigonometric Functions of Acute Angles
16. Find the exact value of the six trigonometric functionsof45degreesfunctions of 45 degrees. o45b=145c=2b = 1o45
a = 1
c2 =a2 +b2
c2 =12 +12 =2 ⇒ c = 2
17. π b 1 2 o
4 2 2
sin 45 = sin = = =
c
π a 1
2
2
2
4
cos 45 = cos = = =
c
o
b 1
π a 1
1
1
4
tan45 = tan = = =
a
π o 1
1
4
cot 45 = cot = = =
b
o
2
1
2
4
csc 45 = csc = = =
b
π c o
2
1
2
4
sec 45 = sec = = =
a
π c o
18. Find the exact value of the six trigonometric functionsof30and60degreesfunctions of 30 and 60 degrees. o30o30o302o30b22b2o60o60
a a
2a = 2 so a = 1
2 2 2
2b22 == 1222 +−1b22= 4 −1 = 3
c2 = a2 +b2 bb = 33
19. π a 1
π b 3
2
6
sin 30 = sin = =
c
o
2
6
cos 30 = cos = =
c
o
sin30o 3
3
3
1
3 2
1 2
cos30
sin30
6
tan30 = tan = = = = o
o π
3
1 3
1
tan 30
1
6
cot 30 = cot = = = o
o π
2
1 2
1
sin 30
1
6
csc 30 = csc = = = o
o π
2 3
2
1
1
sec 30 = sec = = = = o
o π
6
cos 30
3 2
3
3
20. i 60 i π b 3 o
π a 1
o
3 2
sin = sin = =
c
2
3
cos 60 = cos = =
c
i 60o 3 2
3
3
1
1 2
cos60
sin60
3
tan60 = tan = = = = o
o
o π
3
3
3
1
tan 60
1
3
cot 60 = cot = = = o
o π
2 3
2
1
1
csc60 = csc = = = = o
o π
3
sin60
3 2
3
3
o = = 1 = 1 = π
2
3 cos 60 1 2
sec 60 sec o
