Help with trig half angles

[Someone109]

The problem says to find sin of θ /2 with cos of θ being (-√ 5)/3 and 180<θ<270

I got (√1-(√5/3))/√2, but I can't seem to simplify it.

Oh never mind I've found the problem

 

[Deleted member 4993]

The problem says to find sin of θ /2 with cos of θ being (-√ 5)/3 and 180<θ<270

I got (√1-(√5/3))/√2, but I can't seem to simplify it.

Your answer is incorrect.

What equation did you use to arrive at that answer?

Please show intermediate calculations.

 

[Otis]

… I got √(1-(√5/3))/√2

Hi Someone109. That result contains a sign error. A typo, perhaps.

By the way, we don't need grouping symbols around the numerator (above) because a radical sign is a grouping symbol. However, we do need grouping symbols around the radicand -- to show what's inside the radical sign. Like this:

√[1 - (-√5/3)] / √2

… can't seem to simplify it. Oh never mind I've found the problem

Oh, good. I assumed that "simplify it" means rationalizing the denominator:

(√3 + √15) / 6

?

 

[Deleted member 4993]

Hi Someone109. That result contains a sign error. A typo, perhaps.

By the way, we don't need grouping symbols around the numerator (above) because a radical sign is a grouping symbol. However, we do need grouping symbols around the radicand -- to show what's inside the radical sign. Like this:

√[1 - (-√5/3)] / √2

Oh, good. I assumed that "simplify it" means rationalizing the denominator:

(√3 + √15) / 6

?

otis,

please check your math - the last part

 

[Otis]

please check your math - the last part

Hi Subhotosh. I checked, but I couldn't find an issue.

Let θ = 221.8103º (rounded)

cos(θ) ≈ -0.745356

That's -√(5)/3 rounded.

sin(θ/2) ≈ 0.934172

That's (√3 + √15)/6 rounded.

?

 

[Deleted member 4993]

Hi Subhotosh. I checked, but I couldn't find an issue.

Let θ = 221.8103º (rounded)

cos(θ) ≈ -0.745356

That's -√(5)/3 rounded.

sin(θ/2) ≈ 0.934172

That's (√3 + √15)/6 rounded.

?

Apologies - my answer looked so different

√[1 - (-√5/3)] / √2

= √[1 + √5/3] / √2

= √[(3 + √5)/3] / √2

= √[(3 + √5)/3] / √2

= √[3 + √5] / √6

= √[(3 + √5)] * √6 / 6

= √[(3 + √5) *6]/ 6

= √[(18 + 6√5)]/6 ..................RIGHT NOW THOSE LOOK SO DIFFERENT

=√[(3 + 15 + 2*√3√3√5)]/6

=√(√3 + √15)²/6

=(√3 + √15)/6............................. Going to the corner now..............

 

[Steven G]

Apologies - my answer looked so different

√[1 - (-√5/3)] / √2

= √[1 + √5/3] / √2

= √[(3 + √5)/3] / √2

= √[(3 + √5)/3] / √2

= √[3 + √5] / √6

= √[(3 + √5)] * √6 / 6

= √[(3 + √5) *6]/ 6

= √[(18 + 6√5)]/6 ..................RIGHT NOW THOSE LOOK SO DIFFERENT

=√[(3 + 15 + 2*√3√3√5)]/6

=√(√3 + √15)²/6

=(√3 + √15)/6............................. Going to the corner now..............

And you are our leader. How embarrassing.

 

[Otis]

Apologies - my answer looked so different …

That's okay; appearances have led me astray, a few times.

√(18 + 6√5)/6 is a good answer, too. Eliminating radicals within radicals (if possible) is just a personal preference, when simplifying final answers.

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