# Help With A level Trig how do I find the values of theta for the equation: sin(60∘+θ)=cosθ
for 0∘≤θ≤360∘

Edit: I have got the lowest value which is 15, I am just unsure on how to work out the 2nd value
(edited 2 months ago) Original post by max33456789
how do I find the values of theta for the equation: sin(60∘+θ)=cosθ
for 0∘≤θ≤360∘

Edit: I have got the lowest value which is 15, I am just unsure on how to work out the 2nd value

You could use the complementary identity for sin or cos. Original post by mqb2766
You could use the complementary identity for sin or cos.

I used cos theta = sin (90 - theta)
I am unsure how to calculate the other value using it though Original post by max33456789
I used cos theta = sin (90 - theta)
I am unsure how to calculate the other value using it though

Thats what I meant. So when you do arcsin, what are the multiple values? If necessary, think of cast or sin curves. Original post by mqb2766
Thats what I meant. So when you do arcsin, what are the multiple values? If necessary, think of cast or sin curves.

When I done it I done:
sin(60+theta) = sin(90-theta)
60+theta = 90-theta
Then solved to get theta = 15

How do I do the arcsin in this case?

Edit: Do I do sin 15 and then plot against a sin graph to then see where it intersects to give me my 2nd value?
(edited 2 months ago) Original post by max33456789
When I done it I done:
sin(60+theta) = sin(90-theta)
60+theta = 90-theta

Then solved to get theta = 15

How do I do the arcsin in this case?

When you go from line 1 to line 2 (bold) you take inverse sin or arcsin. sin has a restricted domain to make it invertible so when you do the inverse its got multiple solutions. As an example
sin(x) = sin(90-x)
is obviously satisfied by x=45 so sin(x) = 1/sqrt(2) but also by x=225
https://www.desmos.com/calculator/bybfbtk8ci
when sin(x)=-1/sqrt(x).

arcsin is has a range -90 to 90 and you have to explcitly think about solutions outside that domain.
(edited 2 months ago) It would just be easier to use addition formulae. Original post by mqb2766
When you go from line 1 to line 2 (bold) you take inverse sin or arcsin. sin has a restricted domain to make it invertible so when you do the inverse its got multiple solutions. As an example
sin(x) = sin(90-x)
is obviously satisfied by x=45 so sin(x) = 1/sqrt(2) but also by x=225
https://www.desmos.com/calculator/bybfbtk8ci
when sin(x)=-1/sqrt(x).

arcsin is has a range -90 to 90 and you have to explcitly think about solutions outside that domain.

Thank you I have got the right answer now. I had to add 180 each time, which makes sense now I look at the ATSC diagram Original post by max33456789
Thank you I have got the right answer now. I had to add 180 each time, which makes sense now I look at the ATSC diagram

Thats correct, solutions occur every +/-k*180. Its down to the +/- solutions rather than the usual positive or negative symmetry in the cast diagram. As noted above you could do the angle addition identity and that would come out as tan() = ... and obv tan solutions repeat every 180.
(edited 2 months ago)