Calculate sin2a+cos2a+tan2a, if sina/2=1/3 and a is in QI

[Heather]

I need help solving this trig. problem pleaaase
I started it but i'm not sure if i did it correctly...

Calculate sin2a+cos2a+tan2a, if sina/2=1/3 and a is in QI.

I doubled 1/3 to equal 2/3...not sure if that was something i could do but i did it...

so i have sina=2/3, cosa=sqrt of 5/3, and tana=2/sqrt 5

i used the double angle identity to change the equation to

2sina*cosa+1-sin^2a+2tana/1-tan^2a

plugged in the numbers...

2*(2/3)*(sqrt5/3)+1-2*(2/3)^2+[2*(2/sqrt5)]/[1-(2/sqrt5)^2]

i multiplied it out and got 4.086

is this correct?

 

[Loren]

sin(a/2)=1/3 and a is in QI

I would use \(\displaystyle \sin \frac{a}{2}=\sqrt{\frac{1-cos a}{2}}\) to arrive at cos a = 7/9. (unless I made a mistake.)

Then I would sketch the angle and use Pythagorean Thm to determine the other leg. Using the sketch, I would arrive at cos a = 7/9, sin a = whatever and tan a = something else.

I would then determine sin 2a = 2sina cosa arriving at 64/63. In like manner I would determine cos 2a and tan 2a and add them up.

I think the process will get you what you want. I'm not sure that I didn't make a mistake, so you need to do it carefully.

 

[Heather]

ohhh okay i see it now! thank you so much!