5.4, 5.5, 5.8, Trigonometric Functions

How do you get the graph of y=arccosx?

- Restrict the domain of y=cosx to 0 ≤ x ≤ π (so only one value of x for each value of cosx) - The range of y=cosx is still -1 ≤ cosx ≤ 1 - So the domain of y=arccosx is -1 ≤ x ≤ 1 - and the range is 0 ≤ arccosx ≤ π

How do you get the graph of y=arctanx?

- Restrict the domain of y=tanx to -π/2 ≤ x ≤ π/2 (so only one value of x for each value of tanx) - The range of y=tanx is still x ∈ ℝ - So the domain of y=arctanx is x ∈ ℝ - and the range is -π/2 ≤ arctanx ≤ π/2

secx = k and cosecx = k have no solutions for...

-1 < k < 1

What is the relationship between the graphs of the trig functions and inverse trig functions?

-They are a reflection of each other in the line y=x - The range and domain of each are swapped - They run along the opposite axis - Run in opposite direction (i.e. left to right = down to up)

Why do you need to restrict the domain to graph the inverse trig functions?

A function can be one-to-one or many-to-one mappings but can't be one-to-many mappings. Like a calculator, functions must have one output. As the trig functions are many-to-one mappings, when the range and function are swapped for the inverse trig functions, they are one-to-many mappings. The domain must be restricted to make it one-to-one. You only plot the graphs between certain x values, so that for each x value, you end up with one y value. Only one-to-one functions have inverses

Describe the graph of y=arccosx

Domain= -1 ≤ x ≤ 1 Range= -π/2 ≤ arcsinx ≤ π/2 Endpoints = (1, π/2) (-1, -π/2)

Describe the graph y=arcsinx

Domain= -1 ≤ x ≤ 1 Range= -π/2 ≤ arcsinx ≤ π/2 Endpoints = (1, π/2) (-1, -π/2)

Describe the graph of y=arctanθ

Domain= x ∈ ℝ Range= -π/2 ≤ arctanx ≤ π/2 Asymptotes= y=π/2, y=-π/2

Describe the graph of y = secθ

Domain= x ∈ ℝ (x ≠ (2n-1)π/2, n ∈ ℤ) Range= y ≤ 1 ∪ y ≥ 1 Period= 2π Asymptotes= x =(2n-1)π/2, n ∈ ℤ (where cosθ = 0)

Describe the graph of y = cotθ

Domain= x ∈ ℝ (x ≠ nπ, n ∈ ℤ) Range= y ∈ ℝ Period= π Asymptotes= x = nπ, n ∈ ℤ (where sinθ = 0)

Describe the graph of y = cosecθ

Domain= x ∈ ℝ (x ≠ nπ, n ∈ ℤ) Range= y ≤ 1 ∪ y ≥ 1 Period= 2π Asymptotes= x = nπ, n ∈ ℤ (where sinθ = 0)

What are the complementary trig functions?

Each trig function has a complementary trig function. The sin of an angle is equal to the cos 90 - that angle. (because same value if use same sides to form ratio but use different angle) sinθ = cos(90 - θ)

How do you get the graph of y=arcsinx?

Restrict the domain of y=sinx to -π/2 ≤

What does the inverse function of a trig function do?

Reverses the function. If you apply the inverse function on a value of the trig function, you are finding the angle which when sined gets you you input e.g. sinx = 1/2 arcsin(1/2) = 30° arcsin(sinx) = x

What is the relationship between the graphs of tanx and sinx?

The inverses of each look like each other

How do you remember the reciprocal trig functions?

The reciprocal function of each trig function has the starting letter as its third letter sinθ → cosecθ cosθ → secθ tanθ → cotθ

How do you find the graph of y = secθ?

The reciprocal of the y = cosθ graph: - reciprocating preserves sign so stays the same side of the x-axis - A small cosx value reciprocated makes a large secx value (e.g. cosx = 1/2 so secx = 2) - The minima of secx at the maxima of cosx (as 1 reciprocated is 1) (biggest n makes smallest 1/n) - When cosx = 0, sec is undefined as (secx = 1/0) so an asymptote (odd integer multiples of 90°) - The closer to 1 cosx is, the closer to 1 secx is - Each loop takes π

How do you find the graph of y = cosecθ

The reciprocal of the y = sinθ graph: - reciprocating preserves sign so stays the same side of the x-axis - A small sinx value reciprocated makes a large cosecx value (e.g. sinx = 1/2 so cosecx = 2) - The minima of cosecx at the maxima of sinx (as 1 reciprocated is 1) (biggest n makes smallest 1/n) - When sinx = 0, cosec is undefined as (cosecx = 1/0) so an asymptote (integer multiples of 180°) - The closer to 1 sinx is, the closer to 1 cosecx is - Each loop takes π

How do you find the graph of y = cotθ?

The reciprocal of the y = tanθ graph: - reciprocating preserves sign so stays the same side of the x-axis - A small tanx value reciprocated makes a large cotx value (where tanx is almost 0, cotx is almost infinity) - The minima of cotx at the maxima of tanx (biggest n makes smallest 1/n) - When sinx = 0, cotx is undefined as (cotx = cosx/sinx) so an asymptote (integer multiples of 180°) (or tanx = 0) - Where asymptote for tanx, cotx = 0 and vice versa - cotx = 0 for odd integer multiples of 90° (where cosx=0 as 0/sinx = 0) - Same shape as tanx but in opposite direction and asymptotes and cotx=0 are at opposite x values

What are the inverse trig functions?

arcsinx (sin⁻¹x) arccosx arctanx

sec(90 - θ) =

cosecθ

What is cosecant?

cosecθ = 1/sinθ

What are the further trig identities required?

cos²θ + sin²θ ≡ 1 1 + tan²θ ≡ sec²θ (dividing by cos²θ) 1 + cot²θ ≡ cosec²θ (dividing by sin²θ)

sin(90 - θ) =

cosθ

tan(90 - θ) =

cotθ

What is cotangent?

cotθ = 1/tanθ

What is secant?

secθ = 1/cosθ

Write cos50 in terms of sin?

sin40

arcsink = α ↔

sinα = k α is the angle whose value when sined is k k is the value of the angle α when sined

What are the reciprocal trigonometric functions?

the reciprocal of each trig function

What is an inverse function?

y = f(x) f⁻¹[f(x)] = x f⁻¹(y) = x Reverses the function If the inverse function is applied to the output of the function, the input is obtained