a.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» solutions because solutions are in quadrants I and II, and the domain includes one full rotation

b.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» solutions because solutions are in quadrant I and IV, and the domain includes two full rotations

c.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math» solutions because solutions are in quadrants II and IV, and the domain includes one and a half rotations

d.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» solutions because solutions are in quadrants I and IV, and the domain includes one full rotation

a.

d.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»sec«/mi»«mo»§#952;«/mo»«mo»+«/mo»«msqrt»«mn»2«/mn»«/msqrt»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»sec«/mi»«mo»§#952;«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«msqrt»«mn»2«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Recall «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sec«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mn»1«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mfrac»«mo»=«/mo»«mfrac»«msqrt»«mn»2«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math».

The two angles within the first rotation are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mn»3«/mn»«mo»§#960;«/mo»«/mrow»«mn»4«/mn»«/mfrac»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mn»5«/mn»«mo»§#960;«/mo»«/mrow»«mn»4«/mn»«/mfrac»«/mstyle»«/math». Determine any angles coterminal to these within the given domain.


So, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mfrac»«mrow»«mn»3«/mn»«mo»§#960;«/mo»«/mrow»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfrac»«mrow»«mn»3«/mn»«mo»§#960;«/mo»«/mrow»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mfrac»«mrow»«mn»5«/mn»«mo»§#960;«/mo»«/mrow»«mn»4«/mn»«/mfrac»«/mrow»«/mstyle»«/math».

a.

Either both the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values are positive (quadrant I) or they are both negative (quadrant III) for the tan ratio to be positive.

The terminal arm will lie in quadrant I or quadrant III.

In quadrant I,


In quadrant III,

b.

The value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» must be negative for the cosine ratio to be negative. The terminal arm will lie in quadrant II or quadrant III.

In quadrant II,


In quadrant III,

c.

The sine ratio is positive, so the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-value must be positive. The terminal arm will lie in quadrants I or quadrant II.

In quadrant I,


In quadrant II,

d.

The cotangent ratio will be positive in quadrant I or quadrant III.

In quadrant I,


In quadrant III,

a.

Isolate the trigonometric ratio.


Note the interval, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»§#8804;«/mo»«mo»§#952;«/mo»«mo»§#60;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math». It permits one full positive rotation, but does not include «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math».

The trigonometric ratio «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» indicates that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». This occurs at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#960;«/mo»«/mstyle»«/math».

b.

Isolate the trigonometric ratio.


Note the interval. Values in the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mo»§#8804;«/mo»«mo»§#952;«/mo»«mo»§#8804;«/mo»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math» are acceptable.

So, a half rotation in the negative direction through quadrants III and IV and a full rotation in the positive direction are permitted.

The trigonometric ratio «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»tan«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math» indicates the reference angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mstyle»«/math».

Also, the terminal arm will lie in the quadrants where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» have opposite signs. The possible quadrants are II and IV.

In quadrant II, the angle of rotation is


In quadrant IV, the principal angle is


By checking for coterminal angles between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mo»§#960;«/mo»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math», it can be seen that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math» is also a solution.

c.

Isolate the trigonometric ratio.


The interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» allows for a full positive rotation passing through all four quadrants and it also allows for a full negative rotation through all four quadrants.

The ratio «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«msqrt»«mn»2«/mn»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math» indicates the reference angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»45«/mn»«mo»§#176;«/mo»«/mstyle»«/math».

The ratio is positive, which means the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-value is positive and the terminal arm lies in quadrant I or quadrant II.

If the terminal arm is in quadrant I, the principal angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»45«/mn»«mo»§#176;«/mo»«/mstyle»«/math» and the negative coterminal angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«mo»+«/mo»«mn»45«/mn»«mo»§#176;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»315«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

If the terminal arm is in quadrant II, the principal angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»180«/mn»«mo»§#176;«/mo»«mo»§#8722;«/mo»«mn»45«/mn»«mo»§#176;«/mo»«mo»=«/mo»«mn»135«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» and the negative coterminal angle would be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»135«/mn»«mo»§#176;«/mo»«mo»§#8722;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»225«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

d.

Collect like terms and isolate the trigonometric ratio.


The sine ratio is negative where the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values are negative. The terminal arm lies in quadrant III or quadrant IV.

The reference angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»30«/mn»«mo»§#176;«/mo»«/mstyle»«/math».

The interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»180«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» allows for one full negative rotation through all four quadrants and one half positive rotation through quadrants I and II. There are no possible solutions in quadrant I or quadrant II, so the only possible solutions are negative.

In quadrant III,


In quadrant IV, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«msup»«mn»30«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«/mrow»«/mstyle»«/math».

e.

Collect like terms and isolate the trigonometric ratio.


Because the cotangent ratio is the reciprocal ratio to tangent, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»tan«/mi»«mi»x«/mi»«mo»=«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».

The ratio indicates the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values are opposite in sign and the terminal arm lies in quadrant II or quadrant IV. The reference angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»45«/mn»«mo»§#176;«/mo»«/mstyle»«/math». The interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mn»180«/mn»«mo»§#176;«/mo»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»360«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» allows for a full positive rotation through all four quadrants and half of a negative rotation through quadrants III and IV.

In quadrant II, the positive solution is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»180«/mn»«mo»§#176;«/mo»«mo»§#8722;«/mo»«mn»45«/mn»«mo»§#176;«/mo»«mo»=«/mo»«mn»135«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math». A negative rotation does not pass into this quadrant within the given interval.

In quadrant IV, the positive solution is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»360«/mn»«mo»§#176;«/mo»«mo»§#8722;«/mo»«mn»45«/mn»«mo»§#176;«/mo»«mo»=«/mo»«mn»315«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

The negative solution in quadrant IV is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»45«/mn»«mo»§#176;«/mo»«/mstyle»«/math».

f.

Isolate the trigonometric ratio.


The secant ratio is the reciprocal ratio of cosine.

Because the secant ratio is negative, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-value is negative. The terminal arm lies in quadrant II or quadrant III.

The interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mo»§#8804;«/mo»«mo»§#952;«/mo»«mo»§#8804;«/mo»«mn»3«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math» allows for a full rotation and a half in the positive direction. All four quadrants will be passed through, and quadrants I and II will be passed through a second time.

A half negative rotation also occurs, with quadrants III and IV being passed through.

The reference angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mstyle»«/math».

The positive angle in quadrant II is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#960;«/mo»«mo»§#8722;«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»5«/mn»«mo»§#960;«/mo»«/mrow»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math».

A positive coterminal angle occurs in quadrant II at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«mo»+«/mo»«mfrac»«mrow»«mn»5«/mn»«mo»§#960;«/mo»«/mrow»«mn»6«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»17«/mn»«mo»§#960;«/mo»«/mrow»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math».

In quadrant III, there is a positive angle of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#960;«/mo»«mo»+«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»6«/mn»«mo»§#960;«/mo»«mo»+«/mo»«mo»§#960;«/mo»«/mrow»«mn»6«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»7«/mn»«mo»§#960;«/mo»«/mrow»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math».

There is a negative angle in quadrant III of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#8722;«/mo»«mo»§#960;«/mo»«mo»+«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»5«/mn»«mo»§#960;«/mo»«/mrow»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math».

a.

b.

Factor first.



Because the tangent ratio is positive, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values must have the same sign.

The terminal arm lies in quadrant I or quadrant III.

The reference angle is


The solution in quadrant I is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»§#8784;«/mo»«mn»63«/mn»«mi mathvariant=¨normal¨».«/mi»«msup»«mn»43«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«/mrow»«/mstyle»«/math».

The solution in quadrant III is


Also,


Now, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values must be of the opposite signs, and the terminal arm lies in quadrant II or quadrant IV.

The reference angle is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»45«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

In quadrant II, the solution is


In quadrant IV, the solution is

c.

Factor first.


or


The interval is given as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mstyle»«/math», which means the angle can be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» radians and up to, but not including, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math» radians.

The sine ratio involves «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values and if «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-value must be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math». The terminal arm lies on the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis.

So, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mo»§#960;«/mo»«/mrow»«/mstyle»«/math».

Next, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» indicates «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math», which means the terminal arm lies on the positive «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis.

So, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math».

d.

Factor first.


or


The secant ratio is the reciprocal ratio for cosine. The cosine ratio is positive where the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-values are positive, in quadrants I and IV.


The interval is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨»«mrow»«mo»§#8722;«/mo»«mn»180«/mn»«mo»§#176;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mn»180«/mn»«/mrow»«/mfenced»«/mstyle»«/math», which means a negative rotation is allowed through quadrants III and IV and a positive rotation through quadrants I and II is allowed.

The solution in quadrant I is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mn»70«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»53«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

The solution in quadrant IV is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»70«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»53«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

Also,


which means


The terminal arm must lie on the negative «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis.

The solution is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mo»§#8722;«/mo»«mn»180«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».

Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»270«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» into the equation and check to see if «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»LS«/mi»«mo»=«/mo»«mi»RS«/mi»«/mrow»«/mstyle»«/math».


Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»180«/mn»«mo»§#176;«/mo»«/mstyle»«/math» into the equation and check to see if «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»LS«/mi»«mo»=«/mo»«mi»RS«/mi»«/mrow»«/mstyle»«/math».


Therefore, the solution «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mn»270«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» is a solution and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»§#952;«/mo»«mo»=«/mo»«mn»180«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» is not.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» does not have a solution.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»§#8800;«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math» because «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mo»§#952;«/mo»«mo»=«/mo»«mi»y«/mi»«/mrow»«/mstyle»«/math». On the unit circle, the leg length, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math», is always less than the hypotenuse of length «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», so «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»§#8804;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».

a.

Also,


and


so,