Evaluation of an expression under conditions

Question

Sin(x-y); tan(x)=5/12, x in quadrant III, sin(y)= -3sqrt10/10, y in quadrant IV

Answer 1

Sin(x-y); tan(x)=5/12, x in quadrant III, sin(y)= -3sqrt10/10, y in quadrant IV

I say sin(x)=-5/(5²+12²)^(1/2), and cos(x)=-12/(5²+12²)^(1/2)

^{I say cos(y)=(1-30/100)(1/2) -- positive square root since looking at cos in 4th quadrant}

^{and sin(x-y)=sin(x)*cos(y)-sin(y)-cos(x)}

Answer 2

To start, there is a trig identity for the sine function that is as follows:

sin(a - b) = sin(a)cos(b) - cos(a)sin(b).

So, each of those 4 pieces of that equation will NEED to be determined.

Here are a few more trig identities that will be utilized throughout the solution:

sin²(θ) + cos²(θ) = 1

1 + tan²(θ) = sec²(θ)

Utilizing the third out of the 3 equations I've presented, let's use the fact of tan(x) = 5/12 first.

1 + (5/12)² = sec²(x) 1 + 25/144 = sec²(x) 169/144 = sec²(x) cos²(x) = 144/169

cos(x) = ±12/13

Now, the square-root does necessarily require us to assign a plus-or-minus to right side of the equation. However, the other fact you were given helps you determine whether cos(x) is the negative or positive result. x is located in quadrant III. This quadrant let's you know that the cosine or sine of angles in this quadrant WILL be negative.

Mathematically said, say some angle a = 5π/4:

cos(a) = -√(2)/2 sin(a) = -√(2)/2

So, cos(x) = -12/13.

sin²(x) + cos²(x) = 1 sin²(x) + 144/169 = 169/169 sin²(x) = 25/169

sin(x) = -5/13.

We already know that sin(y) = -3√(10)/10.

So:

sin²(y) + cos²(y) = 1 9/10 + cos²(y) = 1 cos²(y) = 1/10 cos(y) = ±√(10)/10

Similarly to x, since we were told that y is in Quadrant IV. Quadrant IV has it that:

sin(y) = – # cos(y) = + #

So, cos(y) = +√(10)/10.

So:

sin(x-y) = sin(x)cos(y) - cos(x)sin(y) = (-5/13)(√(10)/10) - (-3√(10)/10)(-12/13) = -5√(10)/130 - 26√(10)/130

Therefore:

sin(x-y) = -31√(10)/130.

As a double-check: since y is in Quadrant IV and x is in Quadrant III, that means that the angle y is greater than the angle x. This means then x-y results in a negative angle AND since QIII and QIV are only different by one quadrant, then their difference will be in QI if the difference was positive and will be in QIV if the difference is negative.

If x-y is in QIV, then the sin(x-y) must be a negative number.

sin(x-y) = -31√(10)/10 checks out with this deduction.