Datrysiad Cyffredinol mewn Radiannau

Datrysiad Cyffredinol mewn Radiannau
General Solution in Radians

​Mathemateg Bellach Uned 4
Further Maths Unit 4

Sin x

PV + 2πn
SV + 2πn

Mae sin yn ailadrodd bob 360^o felly bob 2π radian.
Sine repeats every 360^o so every 2π radians.

Cos x

± PV + 2πn

Mae cos hefyd yn ailadrodd bob 360^o felly bob 2π radian.
Cosine also repeats every 360^o so every 2π radians.

Tan x

PV + πn

Mae tan yn ailadrodd bob 180^o felly bob π radian.
Tan repeats every 180^o so every π radians.

Multiple Choice

Datrysiad cyffredinol sin x = 1/2 yw ...

The general solution of sin x = 1/2 is ...

• x = ± π/6 + 2πn

• x = π/6 + 2πn,

  x = 5π/6 + 2πn

• x = π/6 + πn

• x = π/6 + 2πn,

  x = 11π/6 + 2πn

Match

Parwch y canlynol.

Match the following.

• sin x = √3/2

• cos x = 1/2

• tan x = √3

• sin x = 0

• cos x = 0

x = π/3 + 2πn,

x = 2π/3 + 2πn

x = ± π/3 + 2πn

x = π/3 + πn

x = πn

x = ± π/2 + 2πn

• x = π/3 + 2πn,

  x = 2π/3 + 2πn

• x = ± π/3 + 2πn

• x = π/3 + πn

• x = πn

• x = ± π/2 + 2πn

Onglau Cyfansawdd
Compound Angles

sin (2x) = 1/2

2x = π/6 + 2πn, 2x = 5π/6 + 2πn

x = π/12 + πn, x = 5π/12 + πn

Darganfod y datrysiad cyffredinol ar gyfer 2x.
Find the general solution for 2x.

YNA rhannu trwodd hefo 2.
THEN divide by 2.

Multiple Choice

tan (x/2) = √3

x/2 = π/3 + πn

• x = (2π/3) + 2πn
• x = (π/3) + 2πn
• x = (π/6) + πn/2

• x = (2π/3) + πn/2

Drag and Drop

cos 2x = 1



2x =

Placeholder

x = ​

Placeholder

Drag these tiles and drop them in the correct blank above

2πn

πn

4πn

± π/2 + 2πn

± π/3 + 2πn

± π/4 + πn

± π/6 + πn

Hotspot

Chwiliwch y gwall.
Spot the error.

Multiple Choice

cos (4x) = √2/2

• x = ± π/4 + 2πn

• x = π/16 + πn/2

  x = 15π/16 + πn/2

• x = ± π/16 + πn/2

• x = π/4 + 2πn

  x = 3π/4 + 2πn

Drag and Drop

cos 4x + cos 6x + cos 2x = 0

cos 4x + 2 cos (6x+2x)/2 cos (6x-2x)/2 = 0

cos 4x + 2 cos

Placeholder

cos​

Placeholder

= 0

cos 4x (1 + cos 2x) = 0

​cos 4x = 0 and cos 2x = -1

​

Placeholder

and ​ 2x = πn

​

Placeholder

and ​

Placeholder

Drag these tiles and drop them in the correct blank above

4x

2x

8x

x

4x = ± π/2 + 2πn

x = ± π/8 + πn/2

x = πn/2

x = 2πn

x = ± π/16 + πn/2

4x = ± π/2 + nπ