Question

Trigonometry

Angle Sum and Difference, Double Angle and Half Angle Formulas
Find the exact value of:
If sinx = 2/8, find Cos (3x)

Answer 1

3√15/4

Explanation:

sin x = 2/8

• from the trigonometric rule sin²x+cos²x = 1
• cos²x = 1-sin²x

sinx = 2/8

sin²x = (2/8)² = 2²/8² = 4/64

1- sin²x = 1-4/64 = 64 -4/64 = 60/64 = 15/16

• cos²x = 15/16

cos x = √(15/16) = √15 /4

• cos 3x

3 x cos x = 3 x √15/4 = 3√15/4

Answer 2

`cos3x=(3√(15))/(16)`

Explanation:

Notes that:

`sinx=(2)/(8)=(1)/(4)`.

We use sinx to find cosx:

`cos^2x=1-sin^2x=1-((1)/(4))^2=(15)/(16)`, then

`cosx=(√(15))/(4)`.

Now we use formule for double angle of sin and cos, to find cos2x and sin2x:

`sin2x=2sinxcosx=2*(1)/(4)*(√(15))/(4)=(√(15))/(8)`

`cos2x=cos^2x-sin^2x=(15)/(16)-(1)/(16)=(14)/(16)=(7)/(8)`

Now, we can find cos3x:

`cos3x=cos(2x+x)=cos2x*cosx-sin2x*sinx=`

`(7)/(8)*(√(15))/(4)-(√(15))/(8)*(1)/(4)`

`(6√(15))/(32)=(3√(15))/(16)`