Please help me understand the relationship between this question and solution

[bushra1175]

Hi everyone. Question a) is asking which quadrants sine theta = cos theta lie in, however the solution is dividing sine theta with cos theta to equate to something else, rather than their values equating to each other (which I don't think is possible). So how is it this solution is acceptable?

 

[joshuaa]

when [MATH]\theta = 45°= \frac{\pi}{4}[/MATH] (first quadrant)

[MATH]\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}[/MATH]
when [MATH]\theta = 225° = \frac{5\pi}{4}[/MATH] (third quadrant)

[MATH]\sin \frac{5\pi}{4} = \cos \frac{5\pi}{4} = \frac{-\sqrt{2}}{2}[/MATH]
division means
when they both positive [MATH]\frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 > 0[/MATH]
when they both negative [MATH]\frac{-\sin \frac{\pi}{4}}{-\cos \frac{\pi}{4}} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 > 0[/MATH]

 

[JeffM]

This is really about understanding the signs of the trigonometric functions.

The sine and cosine are both positive in the first quadrant and both negative in the third quadrant. Therefore they can be equal only in those quadrants.

 

[Steven G]

Where are they changing the equation sinθ = cosθ ?
Obviously for sinθ = cosθ, both sinθ and cosθ must be of the same sign. This happens in quad I and III.

But what is the angle? One way to find it is to divide both sides by cosθ to get tanθ = 1. In quad I that happens when θ = 45^o and in quadrant III, θ = 225^o.
Any questions??

 

[bushra1175]

Where are they changing the equation sinθ = cosθ ?
Obviously for sinθ = cosθ, both sinθ and cosθ must be of the same sign. This happens in quad I and III.

But what is the angle? One way to find it is to divide both sides by cosθ to get tanθ = 1. In quad I that happens when θ = 45^o and in quadrant III, θ = 225^o.
Any questions??

Sorry if my question is unclear. The solution said sineθ/cosθ = tanθ and I was just thinking, what does sineθ being divided by cosθ have to do with the question where sineθ = cosθ. Joshua kindly explained it above and now I understand

 

[Steven G]

Sorry if my question is unclear. The solution said sinex/cosx = tanx and I was just thinking, what does sinex being divided by cosx have to do with the question where sinex = cosx. Joshua kindly explained it above and now I understand

They are equivalent equation. You are basically asking if 2x+3 = 9 then what does 2x = 6 have to do with the original question. If you divide, multiply, add or subtract the same to both sides you (almost always) get an equivalent equation. Two equations are equivalent if they both have the same solutions.