Download Trigonometric Functions of right triangle 3 2

Trigonometric Functions of right triangle The Trigonometric Ratios opposite b hypotenuse
a
c
θ adjacent
sin θ =
adj. c
opp. b
= cos θ =
= hyp. a
hyp. a
tan θ =
opp . b
adj . c
= cot θ =
= adj . c
opp. b
sec θ =
hyp. a
hyp. a
= csc θ =
= adj . c
opp. b
Exp: Find the six trigonometric ratios of the angle in the following figure. 3
2 θ sin θ =
tan θ =
sec θ =
5 5
2
cos θ =
3
3
2
5
3
5
cot θ =
csc θ =
5
2
3
2
Note: cot θ =
1
1
1
sec θ =
csc θ =
tan θ
cos θ
sin θ
Special Triangles θ in degrees θ in radiuss sin θ
cos θ
tan θ 30o π
1
2 3
2 3
3 45o π
2
2 1 1 60o π
3
2 2
2 1
2 3
3
3 6 4 3 cot θ
3 sec θ
csc θ
2 3
3 2 2 2 2 2 3
3 Fundamental Identities 1. tan θ =
sin θ
cos 2 θ
2. cot θ =
cos θ
sin θ
3. sin 2 θ + cos 2 θ = 1 4. tan 2 θ + 1 = sec 2 θ 5. 1 + cot 2 θ = csc 2 θ Trigonometric functions of Real numbers Def: Let t be any real number and let P(x, y) be the terminal point on the unit circle determined by θ . We define sin ϑ = y cos θ = x tan θ =
csc θ =
y
( x ≠ 0) x
1
x
1
sec θ = ( x ≠ 0) cot θ = ( y ≠ 0) x
y
y
Even‐odd Properties sin( −t ) = − sin t cos( − t ) = cos t tan( −t ) = − tan t csc( −t ) = − csc t sec( −t ) = sec t cot( − t ) = − cot t Trigonometric Graphs The functions sine and cosine have period 2π:
The functions tangent and cotangent have period π:
The functions cosecant and secant have period 2π:
Addition and Subtraction Formulas (Trigonometric Function) Formulas of sine: sin( s + t ) = sin s cos t + cos s sin t sin( s − t ) = sin s cos t − cos s sin t Formulas of cosine: cos( s + t ) = cos s cos t − sin s sin t cos( s − t ) = cos s cos t + sin s sin t Exp: If f ( x ) = sin x , show that f ( x + h) − f ( x)
cosh − 1
sinh
+ cos x
= sin x
h
h
h
Sol: f ( x + h) − f ( x ) sin( x + h) − sin x sin x cosh + cos x sinh − sin x sin x (cosh − 1) + cos x sinh
=
=
=
h
h
h
h
cosh − 1
sinh
= sin x (
) + cos x
h
h
Exercise: Let g ( x ) = cos x , show that
f ( x + h) − f ( x)
1 − cosh
sinh
= − cos x (
) − sin x (
) h
h
h
Formulas for tangent: tan( s + t ) =
tan s + tan t
1 − tan s tan t
tan( s − t ) =
tan s − tan t
1 + tan s tan t
Double‐Angles Formulas sin 2 x = 2 sin x cos x cos 2 x = cos 2 x − sin 2 x = 1 − 2 sin 2 x = 2 cos 2 x − 1 tan 2 x =
2 tan x
1 − tan 2 x
Formulas for Lowering Powers sin 2 x =
1 − cos 2 x
1 + cos 2 x
2
cos x =
2
2
tan 2 x =
1 − cos 2 x
1 + cos 2 x
Exp: Express sin 2 x cos 2 x in terms of the first power of cosine. Sol: sin 2 x cos 2 x =
1 − cos 2 x 1 + cos 2 x 1
1 1 1 + cos 4 x
1 cos 4 x
)= −
⋅
= (1 − cos 2 2 x ) = − (
2
2
4
4 4
2
8
8
Product‐Sum formulas sin u cos v =
1
[sin( u + v ) + sin( u − v )] 2
cos u cos v =
1
[cos( u + v ) + cos( u − v )] 2
sin u sin v =
1
[cos(u − v) − cos(u + v)] 2
Exp: Express sin 3 x sin 5 x as a sum of trigonometric functions. Sol: sin 3 x sin 5 x =
1
[cos( −2 x ) − cos(8 x )] = 1 (cos 2 x − cos 8 x ) 2
2
Exercise: Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine. 1. sin 4 x =
3 1
cos 4 x
− cos 2 x +
8 2
8
3 1
cos 4 x
2. cos 4 x = + cos 2 x +
8 2
8
Exercise: Write the product as a sum 1. sin 2 x cos 3 x =
1
1
(sin 5 x + sin( − x )) = (sin 5 x − sin x ) 2
2
2. cos 5 x cos 3 x =
1
(cos 8 x + cos 2 x ) 2
One‐to‐One Functions Def: A function with domain A is called a one‐to‐one function if no two elements of A have the same image, that is, f ( x1 ) ≠ f ( x 2 ) whenever x1 ≠ x 2 . Horizontal Line Test A function is one‐to‐one if and only if no horizontal line intersects its graph more than once. The inverse of a Function Def: Let f be a one‐to‐one function with domain A and range B. Then its inverse function f −1 has domain B and range A and is defined by f −1 ( y ) = x ⇔ f ( x) = y for any y ← B . −1
Note: f ( x) ≠
1
1
= ( f ( x)) −1 f ( x)
f ( x)
Exp: If f (1) = 5 , f (3) = 7 and f (8) = − 10 , find f
Sol: f
−1
(5) = 1, f
−1
(7) = 3, f
−1
−1
(5), f
−1
(7), f
−1
(−10). (−10) = 8 . Inverse function Property Let f be a one‐to‐one function with domain A and range B. The inverse function f
−1
satisfies f −1 ( f ( x)) = x
f ( f −1 ( x)) = x
∀x ∈ A ∀x ∈ B How to find the Inverse function 1. Write y = f (x) . 2. Solves this equation for x in terms of y (if possible) 3. Interchange x and y. The resulting equation is y = f
Exp: Find the inverse of the function f ( x) = 3x − 2 Sol: 1st. y = 3x − 2 2nd. 3x = y + 2 x =
3rd. y =
Ans: f
−1
x+2
3
( x) =
x+2
3 y+2
3
−1
( x) Exp: Let f ( x ) = x − 2 . Find f
Sol: 1st. y = x − 2
−1
( x) y ≥0 2nd. y 2 = x − 2 x = 2 + y 2 3rd. y = 2 + x 2 x ≥ 0 Thus, f −1 ( x) = x 2 + 2
x ≥ 0 Exercise: Find the inverse function of f. 1. f ( x) = 2 x + 1 Sol : f −1 ( x) =
2. f ( x ) =
1
Sol : f
x+2 3. f ( x) = 4 − x 2
−1
x −1
2 ( x) =
1 − 2x
( x ≠ −2 ) x
x ≥ 0 Sol : f
−1
( x) = 4 − x Inverse Trigonometric Functions The inverse sine function is the function sin −1 with domain [− 1,1] and range ⎡ π π⎤
−1
⎢− 2 , 2 ⎥ defined by sin x = y ⇔ sin y = x . ⎦
⎣
The inverse sine function is also called arcsine, denoted by arcsin. The inverse cosine function is the function cos −1 with domain [− 1,1] and range [0, π ] defined by cos −1 x = y ⇔ cos y = x . The inverse cosine function is called arccosine, denoted by arcos. π
1 π
1
Exp: (1) sin −1 ( ) = sin −1 (− ) = − 2
6
2
6
(2) cos −1 (
3 π
π
) = cos −1 0 = 2
6
2
The inverse tangent function is the function tan −1 with domain R and range ⎡ π π⎤
−1
⎢− 2 , 2 ⎥ defined by tan x = y ⇔ tan y = x . ⎦
⎣
The inverse tangent function is also called arctangent, denoted by arctan. Exp: tan −1 1 =
π
π
tan −1 3 = 4
3
Exercise: (1) cos −1
1 π
= 2 3
3 π
= 2
3
−1
(3) tan 0 = 0 (2) sin −1