Find the angle $ heta $ in the unit circle where the sum of $ sin( heta) $ and $ cos( heta) $ equals 1.5

To find the angle $ \theta $ where the sum of $ \sin(\theta) $ and $ \cos(\theta) $ equals 1.5, we start with the equation:

$ \sin(\theta) + \cos(\theta) = 1.5 $

We can use the Pythagorean identity:

$ \sin^2(\theta) + \cos^2(\theta) = 1 $

Let’s square both sides of the original equation:

$ (\sin(\theta) + \cos(\theta))^2 = 1.5^2 $

This gives us:

$ \sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = 2.25 $

Using the Pythagorean identity:

$ 1 + 2\sin(\theta)\cos(\theta) = 2.25 $

Therefore:

$ 2\sin(\theta)\cos(\theta) = 1.25 $

Which simplifies to:

$ \sin(2\theta) = 1.25 $

However, we know that the range of $ \sin(2\theta) $ is between -1 and 1, so no such $ \theta $ exists.

To find the angle $ heta $ where the sum of $ sin( heta) $ and $ cos( heta) $ equals 1.5, we start with the equation:

$ sin( heta) + cos( heta) = 1.5 $

We can use the Pythagorean identity:

$ sin^2( heta) + cos^2( heta) = 1 $

Squaring both sides of the original equation:

$ (sin( heta) + cos( heta))^2 = 1.5^2 $

This gives:

$ sin^2( heta) + 2sin( heta)cos( heta) + cos^2( heta) = 2.25 $

Using the Pythagorean identity:

$ 1 + 2sin( heta)cos( heta) = 2.25 $

So:

$ 2sin( heta)cos( heta) = 1.25 $

Which simplifies to:

$ sin(2 heta) = 1.25 $

No such $ heta $ exists since $ sin(2 heta) $ is limited to values between -1 and 1.

To find the angle $ heta $ where the sum of $ sin( heta) $ and $ cos( heta) $ equals 1.5:

$ sin( heta) + cos( heta) = 1.5 $

Square both sides:

$ (sin( heta) + cos( heta))^2 = 2.25 $

Using Pythagorean identity:

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