Question

the wind changes again and the weathervane turns another 70°. now it points directly south.

Explanation:

Step1: Analyze the initial direction

Assume the weathervane was initially pointing to direction \( T \). The angle between \( N \) and \( E \) is \( 90^\circ \), and the angle between \( N \) and \( T \) is \( 110^\circ \), so the angle between \( E \) and \( T \) is \( 110^\circ - 90^\circ= 20^\circ \) (east of south? Wait, no, let's re - examine. Wait, the final direction is South. The weathervane turned \( 70^\circ \) to get to South. Let's find the initial direction. Let the initial direction be \( \theta \) from some axis. The angle between the initial direction (before the last \( 70^\circ \) turn) and South is \( 70^\circ \). Also, from the diagram, the angle between North and the initial direction (maybe \( T \)) is \( 110^\circ \). Since North and South are \( 180^\circ \) apart, \( 180^\circ- 110^\circ = 70^\circ \), which matches the turn angle. Wait, maybe the question is to find the initial direction before the \( 70^\circ \) turn. Let's see, if after turning \( 70^\circ \) it points South, then the initial direction (let's call it \( T \)) makes a \( 70^\circ \) angle with South towards East (since in the diagram, \( T \) is to the east of South with a \( 70^\circ \) angle between \( T \) and South? Wait, no, the angle between \( T \) and South is \( 70^\circ \), so the initial direction \( T \) is \( 70^\circ \) east of south? Wait, maybe the question is to find the measure of the angle or the initial direction. Wait, maybe the problem is to find the angle between the initial direction (before the \( 70^\circ \) turn) and South, which is \( 70^\circ \), or the angle between North and the initial direction, which is \( 180 - 70=110^\circ \), which is shown in the diagram. Alternatively, maybe the question is to find where the weathervane was pointing before the last \( 70^\circ \) turn. Since it turned \( 70^\circ \) to point South, and the angle between \( T \) (initial) and South is \( 70^\circ \), and \( T \) is in the fourth quadrant (southeast direction? Wait, no, from the diagram, \( T \) is to the right of the South - East axis? Wait, maybe the key is that the angle between the initial direction ( \( T \)) and South is \( 70^\circ \), so the initial direction is \( 70^\circ \) north of east? No, wait North and South are vertical. Let's use the fact that when you turn \( 70^\circ \) to reach South, the initial direction ( \( T \)) has an angle of \( 70^\circ \) with South, so the direction of \( T \) is \( 70^\circ \) east of south? Wait, no, if you are at \( T \) and turn \( 70^\circ \) towards South (i.e., towards the South axis), you end up at South. So the angle between \( T \) and South is \( 70^\circ \), so \( T \) is \( 70^\circ \) east of south? Wait, maybe the question is to confirm the angle, and the answer is related to the \( 70^\circ \) turn and the \( 110^\circ \) angle. Wait, maybe the problem is to find the measure of the angle between the initial direction (before the \( 70^\circ \) turn) and North, which is \( 110^\circ \), or the angle of the turn. Since the weathervane turned \( 70^\circ \) to go from \( T \) to South, and the angle between North and \( T \) is \( 110^\circ \), and \( 180 - 70=110 \), that checks out. So if we assume the question is about the angle of the turn or the initial direction, and from the diagram, the angle between North and the initial direction ( \( T \)) is \( 110^\circ \), and the turn angle to South is \( 70^\circ \), and \( 110 + 70=180 \) (since North to South is \( 180^\circ \)). So maybe the answer is that the weathervane was pointing \( 70^\circ \) east of south (or \( 20^\circ \) south of east? Wait, no, let's calculate the angle with the east - west axis. The angle between \( T \) and the south axis is \( 70^\circ \), so the angle between \( T \) and the east axis: since south and east are \( 90^\circ \) apart, the angle between \( T \) and east is \( 90 - 70 = 20^\circ \) south of east. Wait, \( 90^\circ-70^\circ = 20^\circ \), so \( T \) is \( 20^\circ \) south of east? Or \( 70^\circ \) east of south? Let's use trigonometry. If we consider the south axis as the y - axis (negative y - axis) and east as the positive x - axis. The direction \( T \) has an angle of \( 70^\circ \) from the south axis (negative y - axis) towards the east axis (positive x - axis). So in standard position (from positive x - axis, counter - clockwise), the angle would be \( 180^\circ+ 20^\circ=200^\circ \)? No, wait, south is \( 270^\circ \) in standard position (if we take east as \( 0^\circ \), counter - clockwise). Wait, no, standard position is from positive x - axis (east) counter - clockwise. So south is \( 270^\circ \), east is \( 0^\circ \), north is \( 90^\circ \), west is \( 180^\circ \). If a direction is \( \theta \) degrees east of south, its angle in standard position is \( 270^\circ-\theta \). If \( \theta = 70^\circ \), then the angle is \( 270 - 70=200^\circ \). The angle between north ( \( 90^\circ \)) and this direction is \( 200 - 90 = 110^\circ \), which matches the diagram. So the initial direction (before the \( 70^\circ \) turn) is \( 70^\circ \) east of south (or \( 20^\circ \) south of east, since \( 90 - 70 = 20 \)), and after turning \( 70^\circ \) towards south (i.e., towards the negative y - axis), it points directly south.

Wait, maybe the problem is simpler. The weathervane turned \( 70^\circ \) to go from its original position to South. The angle between its original position and South is \( 70^\circ \), and the angle between its original position and North is \( 180 - 70=110^\circ \), which is shown in the diagram. So if the question is to find how many degrees the weathervane turned from its position (let's say \( T \)) to South, the answer is \( 70^\circ \). Or if it's to find the angle between \( T \) and North, it's \( 110^\circ \).

Assuming the question is to find the measure of the angle of the turn (the \( 70^\circ \) turn) or the angle between the initial direction and North ( \( 110^\circ \)). Since the problem says "the wind changes again and the weathervane turns another \( 70^\circ \). Now it points directly South", maybe we need to find where it was pointing before the \( 70^\circ \) turn. From the diagram, the angle between North and the initial direction ( \( T \)) is \( 110^\circ \), so the initial direction is \( 110^\circ \) from North, and since North to South is \( 180^\circ \), \( 180 - 110 = 70^\circ \), so the angle between the initial direction and South is \( 70^\circ \), which is the turn angle. So the initial direction is \( 70^\circ \) east of south (because it's \( 70^\circ \) towards east from the south axis).

Answer:

If we consider the angle of the turn, it's \( 70^\circ \). If we consider the angle between the initial direction and North, it's \( 110^\circ \). If we consider the direction of the initial position relative to South, it's \( 70^\circ \) east of south. (Since the problem is a bit unclear, but based on the diagram and the information, the most probable answer related to the \( 70^\circ \) turn is that the weathervane turned \( 70^\circ \) to point South, and the initial direction was \( 70^\circ \) east of south (or the angle between initial and North is \( 110^\circ \)). If we take the turn angle as the answer, it's \( 70^\circ \).