Final answer:
To determine cos(w), we use the trigonometric identity sin^2(w) + cos^2(w) = 1. Since sin(w) = 0, cos(w) must be either -1, 0, or 1. Tan(w) being negative indicates that w is in the second quadrant, leading to the conclusion that cos(w) = -1.
Step-by-step explanation:
To evaluate cos(w) without using a calculator given that tan(w) = -4/3 and sin(w) = 0, we can use the trigonometric identity sin^2(w) + cos^2(w) = 1. Since we are given that sin(w) = 0, this simplifies to cos^2(w) = 1. The cosine of an angle can only be -1, 0, or 1 when the sine of the angle is 0. Considering this and the fact that tan(w) = -4/3, which implies that the angle w is in either the second or the fourth quadrant where the values of cosine are negative and positive respectively, we must decide which of these two quadrants the angle w is in. However, since the sine value is zero, it can only occur at 0 or π radians (0 or 180 degrees), where the cosine values are 1 and -1 respectively.
As tan(w) is negative, we deduce that the angle w is in the second quadrant where cosine is negative, hence cos(w) = -1. Therefore, the correct answer is a) -1.