4 Trigonometry

Question 0

Why do we restrict the domain of say y=sinx to -π/2 ≤ χ ≤ π/2

Answer 0

So that it is a one-one function and not a many-one

Question 1

If an inverse function does not occur after a reflection what happened

Answer 1

It has a one-many mapping

It’s not a function so it has to be a one-one function

Question 2

What can the inverse of sinx, cosx, tanx be called?

Answer 2

Arcsinx, arccosx, arctanx, asinx, acosx, atanx

Question 3

How to work out the inverses without a calculator

Answer 3

Using sohcahtoa and 0,30,45,60,90 degrees

Have a look if the graph means it’s a - or a +

Question 4

How to work out sin^-1(-1/√2)

Answer 4

Soh
H = √2
O = 1

The angle you get is 45degrees,
It’s negative so sin(-45) = -1/√2
Sin^-1(-1/√2) =-45

Question 5

How to work out tan^-1(-1/√3)

Answer 5

Toa
O = 1
A= √3

Angle is 30

Tan(-30) = -1/√3

Tan^-1 (-1/√3) =-30

Question 6

Sec^2x and tan^2x

Answer 6

Sec^2x= Tan^2x +1

Question 7

Cosecx and cot

Answer 7

Cosecx =1+ cot

Question 8

Show tanθ/ (1+tan^2θ) = sinθcosθ

Answer 8

Tanθ/sec^2θ = tanθ/1/cos^2θ = tanθ x cos^2θ = sinθ/cosθ x cos^2θ
=sinθcosθ

Question 9

Simplify cosec(π/2 -x)

Answer 9

= 1/sin(π/2 -x). This can be transformed by a reflection and a translation
To 1/cosx =secx

Question 10

Sec^2x =

Answer 10

1+ tan^2x

Question 11

Cosec^2x =

Answer 11

1 + cot^2x

Question 12

Cosec0 =

Answer 12

1/sin0 = 1/0 so impossible

Question 13

To solve these equations

Answer 13

You need to find a way to replace them with tanθ,sinθ,cosθ,

Question 15

Solve tan^2x + secx =1

Answer 15

(Sec^2x -1) + secx = 1
Sec^2x + secx -2 =0
(Secx -1)(secx +2)
Secx =1 or -2
1/Cosx = 1 or -2 
Cosx = 1 or -1/2

Question 16

y= sinx —–> y= asinx

Answer 16

stretch in y direction scale factor a

Question 17

y= sinx —–>y= sinax

Answer 17

stretch in x direction scale factor 1/a

Question 18

y= sinx —–>y=sin (x+a)

Answer 18

translation by (-a/ 0)

Question 19

y= sinx —–>y= (sinx) +a

Answer 19

translation by (0/a)

Question 20

y= sinx —–> -sinx

Answer 20

reflection in x axis

Question 21

y= sinx —–> sin(-x)

Answer 21

reflection in y axis

Question 22

y= tanx ——-> y= tan2x -1

Answer 22

stretch in x direction scale factor 1/2

translation (0/ -1) to y=tan2x -1

Question 23

cos^-1 2x domain? rang?

Answer 23

sketch cos^-1x
x replaced by 2x stretch in x direction scale factor 1/2
(twice as thin)

domain -1/2 x 1/2 range 0 ≤y ≤ π

Question 24

y=sinx —–> y=sin (2x +1)

Answer 24

x replaced by (x+1) translation (-1/0)

x is replaced by 2x its a stretch scale factor 1/2 in x direction

Question 25

y=sinx translation (0/1)

stretch factor 1/3 in y direction

Answer 25

1/3sinx +1/3

Question 26

sin^-1x domain and range?

Answer 26

Domain -1 ≤ x ≤ 1

Range -π/2 ≤ y ≤ π/2

Question 27

cos ^-1x domain and range?

Answer 27

Domain -1 ≤ x ≤ 1

Range 0 ≤ y ≤ π

Question 28

Tan ^-1x domain and range?

Answer 28

Domain - x E ℝ

Range -π/2 ≤ y ≤ π/2

Question 29

Secx Domain and range

Answer 29

Domain x Ε ℝ, x doesn’t equal +-π/2,-+3π/2, -+5π/2
Range y≤ -1 y≥ 1
Period is 2π

Question 30

Cosecx

Answer 30

Domain x ε R, x does not equal 0,-+π, +-2π, -+3π
Range y ≤ -1 y ≥ 1
Period is 2π

Question 31

Cotx

Answer 31

Domain x ε R, x does not equal 0, -+π, -+2π, -+3π
Range y ε R
Period π