Question

If sin(x+y)tan(x)=(3)/(4) in quadrant III and cos(y)=(4)/(5) in quadrant IV, what can you conclude about the values of sin(x) and tan(y)?

Answer 1

sin(x) is negative because it is in quadrant III, and tan(y) is also negative because y is in quadrant IV.

Step-by-step explanation:

If sin(x+y)tan(x) = \(\frac{3}{4}\) in quadrant III and cos(y) = \(\frac{4}{5}\) in quadrant IV, we can deduce certain properties about sin(x) and tan(y). Knowing that all trigonometric functions are positive in quadrant I, sin is positive in quadrants I and II, tan is positive in quadrants I and III, and cos is positive in quadrants I and IV, we can conclude the following:

• sin(x) must be negative since sin is positive in quadrants I and II and we are given that (x+y) is in quadrant III where sin is negative.
• tan(y) must be negative because tan is positive in quadrants I and III, and we are given that y is in quadrant IV where tan is negative.

In conclusion, both sin(x) and tan(y) are negative with sin(x) being in quadrant III and tan(y) being in quadrant IV.