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Problem: 15 Set: Exercises Page: 442Look in your textbook for this problem statement.Step 1Write 165 as a difference of 180 and another angle.

165 = 180 – 15 .

Step 2By Case 4 of the Symmetry Identities,

sin 165 = sin 15

Step 345 and 30 are two common angles that differ by 15 .

sin (15 ) = sin (45 – 30 )

Step 4Use the difference identity for sine.

sin (15 ) = sin 45 cos 30 – cos 45 sin 30

Step 5

Substitute values for the sine and cosine functions.

Step 6Simplify.

Step 7Therefore,

Problem: 17 Set: Exercises Page: 442Look in your textbook for this problem statement.Step 1π /4 and π /6 are two common angles that differ by π /12.

Step 2

Use the difference identity for sine.

Step 3Substitute values for the sine and cosine functions.

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Step 4Simplify.

Problem: 19 Set: Exercises Page: 442Look in your textbook for this problem statement.Step 1Use the opposite angle identity for cosine.

Step 2π /4 and π /6 are two common angles that differ by π /12.

Step 3Use the difference identity for cosine.

Step 4Substitute values for the sine and cosine functions.

Step 5Simplify.

Therefore,

Problem: 21 Set: Exercises Page: 442Look in your textbook for this problem statement.Step 15π /3 and π /4 are common angles whose sum is 23π /12.

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Step 2Use the sum identity for tangent.

Step 3Substitute values for the tangent function.

Step 4

Problem: 23 Set: Exercises Page: 442Look in your textbook for this problem statement.Step 11275 = 7(180 ) + 15 .

By Case 2 of the Symmetry Identities, cosine values of 1275 and 15 are opposite.

sec 1275 = –1/(cos 15 )

Step 245 and 30 are two common angles that differ by 15 .

cos (15 ) = cos (45 – 30 )

Step 3Use the difference identity for cosine.

cos (15 ) = cos 45 cos 30 + sin 45 sin 30

Step 4

Substitute values for the sine and cosine functions.

Step 5Simplify.

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Step 6Therefore,

Step 7

Problem: 25 Set: Exercises Page: 442Look in your textbook for this problem statement.Step 1

By Case 2 of the Symmetry Identities, the values of both sine and cosine change for 113π /12 and 5π /12.

tan 113π /12 = (tan 5π /12)

cot 113π /12 = 1/(tan 5π /12)

Step 2π /4 and π /6 are common angles whose sum is 5π /12.

Step 3Use the sum identity for tangent.

Step 4Substitute values for the tangent function.

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Step 5Therefore,

Step 6

Problem: 27 Set: Exercises Page: 443Look in your textbook for this problem statement.Step 1Use the Pythagorean identity sin2  x + cos2  x = 1.

sin2  x = 1 – cos2  x 

Since  x  is in Quadrant I, the value of sine is positive. Therefore

Step 2Substitute the cosine value.

Step 3Use the Pythagorean identity sin2 y + cos2 y = 1.

sin2 y = 1 – cos2 y 

Since y  is in Quadrant I, the value of sine is positive. Therefore

Step 4Substitute the cosine value.

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Step 5Use the difference identity for cosine.

Step 6Substitute values on the right side.

Step 7Simplify.

Problem: 29 Set: Exercises Page: 443Look in your textbook for this problem statement.Step 1Use the Pythagorean identity

sec2  x = 1 + tan2  x 

Since  x  is in Quadrant I, the value of cosine is positive. Therefore

Step 2Substitute the tangent value.

Step 3Use the reciprocal identity

Step 4Use the Pythagorean identity sin2  x + cos2  x = 1.

sin2  x = 1 – cos2  x 

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Since  x  is in Quadrant I, the value of sine is positive. Therefore

Step 5Substitute the cosine value.

Step 6Use the Pythagorean identity sin2 y + cos2 y = 1.

cos2 y = 1 – sin2 y .

Since y  is in Quadrant I, the value of cosine is positive. Therefore

Step 7Substitute the sine value.

Step 8Use the sum identity for cosine.

Step 9Substitute values on the right side.

Step 10Simplify.

Problem: 31 Set: Exercises Page: 443Look in your textbook for this problem statement.Step 1Use the reciprocal identity

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Step 2Use the Pythagorean identity sin2  x + cos2  x = 1.

cos2  x = 1 – sin2  x 

Since  x  is in Quadrant I, the value of cosine is positive. Therefore

Step 3Substitute the sine value.

Step 4Use the Pythagorean identity

sec2 y = 1 + tan2 y 

Since y  is in Quadrant I, the value of cosine is positive. Therefore

Step 5Substitute the tangent value.

Step 6Use the reciprocal identity

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Step 7Use the Pythagorean identity sin2 y + cos2 y = 1.

sin2 y = 1 – cos2 y 

Since y  is in Quadrant I, the value of sine is positive. Therefore

Step 8Substitute the cosine value.

Step 9

Use the difference identity for cosine.

Step 10Substitute values on the right side.

Step 11Simplify.

Step 12Use the reciprocal identity

Problem: 33 Set: Exercises Page: 443Look in your textbook for this problem statement.Step 1Use the Pythagorean identity sin2  x + cos2  x = 1.

sin2  x = 1 – cos2  x 

Since  x  is in an acute angle (Quadrant I), the value of sine is positive. Therefore

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Step 2Substitute the cosine value.

Step 3Use the Pythagorean identity sin2 y + cos2 y = 1.

sin2 y = 1 – cos2 y 

Since y  is an acute angle (Quadrant I), the value of sine is positive. Therefore

Step 4

Substitute the cosine value.

Step 5Use the sum identity for cosine.

Step 6Substitute values on the right side.

Step 7Simplify.