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Question 16 of 18

beginning of content:

M:N:MC:15

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SAT

Exam

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Additional Topics in Math

Skill

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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?

Select an Answer

straight pi minus x

Correct Answer: 
No

x minus straight pi

Correct Answer: 
Yes

2 straight pi plus x

Correct Answer: 
No

x minus 2 straight pi

Correct Answer: 
No

View Correct Answer

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.