M:N:MC:15

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Additional Topics in Math

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It is given that $sin space x equals a comma$ where x is the radian measure of an angle and $straight pi over 2 less than x less than straight pi.$

If $sin space w equals minus a comma$ which of the following could be the value of w?

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$straight pi minus x$

Correct Answer:

$x minus straight pi$

Correct Answer:

Yes

$2 straight pi plus x$

Correct Answer:

$x minus 2 straight pi$

Correct Answer:

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Choice B is correct. If an angle with radian measure x such that $straight pi over 2 less than x less than straight pi$ is placed in standard position, its terminal side will fall in Quadrant II, and $sin space x equals a$ will be the y-coordinate of the point P where its terminal side intersects the unit circle. If $sin space w equals minus a comma$ then when the angle with radian measure w is placed in standard position, its terminal side will intersect the unit circle at a point with y-coordinate equal to $minus a.$ There are two such points on the unit circle: the reflection of P across the x-axis, which would correspond to an angle with radian measure –x (and also with radian measures... $minus 6 straight pi minus x comma space minus 4 straight pi minus x comma space minus 2 straight pi minus x comma space 2 straight pi minus x comma space 4 straight pi minus x comma space 6 straight pi minus x space... right parenthesis ;$ and the reflection of P through the origin, which would correspond to an angle with radian measure $x minus straight pi$ (and also with radian measures ... $x space minus space 5 straight pi comma space x minus 3 straight pi comma space x plus straight pi comma space x plus 3 straight pi comma space x plus 5 straight pi... right parenthesis.$ Thus, of the choices given, only $x minus straight pi$ could be the value of w.

Choice A is not the correct answer. In general, $sin open parentheses minus x close parentheses equals minus sin space x$ and $sin open parentheses x space plus space straight pi close parentheses equals minus sin space x comma$ so $sin open parentheses straight pi minus straight x close parentheses equals minus sin open parentheses minus x close parentheses equals minus left parenthesis minus sin space x right parenthesis equals sin space x.$ Therefore, $sin open parentheses straight pi minus straight x close parentheses equals a comma$ not $minus a.$

Choice C is not the correct answer. In general, $sin open parentheses 2 straight pi plus x close parentheses equals sin space x comma$ so $sin open parentheses 2 straight pi plus x close parentheses equals a$ , not $minus a.$

Choice D is not the correct answer. In general, $sin open parentheses x minus 2 straight pi close parentheses equals sin space x comma$ so $sin open parentheses x minus 2 straight pi close parentheses equals a comma$ not $minus a.$

Question Difficulty:

hard

Objective:

Students must be fluent with radian measure and have a conceptual understanding of trigonometric relationships.

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Question 16 of 18

M:N:MC:15

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