Find the numerical value of cos(x-y)

Question

If sin x = -6/10, tan y = 4/5

and both angls lie in quad 3

Answer 1

FOR AN EXACT ANSWER:

 

 

Let's get a better idea of the right triangles associated with each angle.

 

 

sin(x) = opposite/hypotenuse = -6/10, so x's opposite side is 6 and its hypotenuse is 10. Solve for the other leg of the triangle using the Pythagorean Theorem:

 

c² = a² + b²

10² = 6² + b²

100 = 36 + b²

64 = b²

8 = b

 

The angle's adjacent side is b = 8. So:

 

sin(x) = -6/10

cos(x) = -8/10        (cosine is negative in Quadrant III)

tan(x) = 6/8        (tangent is positive in Quadrant III)

 

 

tan(y) = opposite/adjacent = 4/5, so y's opposite side is 4 and its adjacent side is 5. Solve for the hypotenuse of the triangle using the Pythagorean Theorem:

 

c² = a² + b²

c² = 4² + 5²

c² = 16 + 25

c² = 41

c = √(41)

 

The triangle's hypotenuse is c = √(41). So:

 

sin(y) = -4/√(41)        (sine is negative in Quadrant III)

cos(y) = -5/√(41)        (cosine is negative in Quadrant III)

tan(y) = 4/5

 

 

Now, you can use the Trigonometric Subtraction Formula for cosine, plugging in the values above:

 

cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

cos(x - y) = (-8/10)(-5/√(41)) + (-6/10)(-4/√(41))

cos(x - y) = 40/(10√(41)) + 24/(10√(41))

cos(x - y) = 20/(5√(41)) + 12/(5√(41))

cos(x - y) = 32/(5√(41))

cos(x - y) = (32√(41)) / (5(41))

cox(x - y) = (32√(41))/205

 

 

 

FOR AN APPROXIMATE ANSWER:

 

 

Solve for x and for y:

 

 

sin(x) = -6/10

x = sin^-1(-6/10)

x ≈ -36.87° = 323.13°

 

That's in Quadrant IV, so we'll need to reflect the angle over the y-axis to find our actual angle (which is in Quadrant III). x = 180° + 36.87° = 216.87°.

 

 

tan(y) = 4/5

y = tan^-1(4/5)

y ≈ 38.66°

 

That's in Quadrant I, so we'll need to reflect the angle over the x-axis and then the y-axis to find our actual angle (which is in Quadrant III). -38.66° is the reflection over the x-axis, so y = 180° + 38.66° = 218.66°.

 

 

Plug those values into the equation and solve:

 

cos(x - y) = cos(216.87° - 218.66°)

cos(x - y) = cos(-1.79°)

cos(x - y) = 0.9995