Prentice Hall Algebra 2

Chapter 14 Trigonometric Identities and Equations > 14-6 Angle Identities > Practice and Problem-Solving Exercises

What is the exact value of sin 105°?

Think	Write
Write the Sine Angle Sum Identity.	sin (A + B close equals sin A cos B + cos A sin B
Find two angles that sum to 105 degrees and have exact sine and cosine values. Let eh equals , 60 degrees and B = 45 ° . b equals , 45 degrees .	sin ( 60 ° + 45 ° ) = sin 60 ° cos 45 ° + cos 60 ° sin 45 ° sine open 60 degrees plus . 45 degrees , close equals . sine , 60 degrees . cosine , 45 degrees plus . cosine , 60 degrees . sine , 45 degrees
Substitute exact values and multiply.	= 3 2 ( 2 2 ) + 1 2 ( 2 2 ) = 6 4 + 2 4 table with 2 rows and 1 column , row1 column 1 , equals , fraction square root of 3 , over 2 end fraction . open , fraction square root of 2 , over 2 end fraction , close . plus , 1 half . open , fraction square root of 2 , over 2 end fraction , close , row2 column 1 , equals , fraction square root of 6 , over 4 end fraction , plus , fraction square root of 2 , over 4 end fraction , end table
Simplify.	= 6 + 2 4 equals . fraction square root of 6 plus square root of 2 , over 4 end fraction

Verify the identity sin ( π 2 + θ ) + sin ( π 2 − θ ) = 2 cos θ . sine . open . pi over 2 , plus theta . close . plus sine . open . pi over 2 , minus theta . close . equals 2 cosine theta . .
Solve tan ( π 2 − θ ) = 1 tangent . open . pi over 2 , minus theta . close . equals 1 for 0 less than or equal to theta less than 2 pi .
Find the exact value of cosine open negative 315 close degrees .
Find the exact value of sine open negative 105 close degrees .

Error Analysis A question on a test asked, “Between 0 and 2 pi , the equation negative cosine theta equals cosine theta has how many solutions?” A student divided each side by cosine theta to get negative 1 equals 1 and concluded that there are no solutions. What mistake did the student make?
Reasoning Use an angle difference identity to show that sin ( π 2 − θ ) = cos θ . sine . open . pi over 2 , minus theta . close . equals cosine theta . .

A Practice

Verify each identity. See Problem 1.

csc ( θ − π 2 ) = − sec θ co-secant . open . theta minus , pi over 2 . close . equals negative secant theta
sec ( θ − π 2 ) = csc θ secant . open . theta minus , pi over 2 . close . equals co-secant theta
cot ( π 2 − θ ) = tan θ co-tangent . open . pi over 2 , minus theta . close . equals tangent theta
csc ( π 2 − θ ) = sec θ co-secant . open . pi over 2 , minus theta . close . equals secant theta
tan ( θ − π 2 ) = − cot θ tangent . open . theta minus , pi over 2 . close . equals negative co-tangent theta
sec ( π 2 − θ ) = csc θ secant . open . pi over 2 , minus theta . close . equals co-secant theta

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