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6.2 Sum, Difference and Double­Angle Identities
It would be logical to think that the following is true, but it is not:
cos (A ­ B) = cos A ­ cos B
In fact, it has been determined that the following is true:
cos (A ­ B) = cos A cos B + sin A sin B
Example: Find the exact value for cos 15o. (Note: 150 does not exist in a 'special triangle', so we must find two other angles from a special triangle that subtract to give 150.)
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Similarly, there are sum and difference identities for sine, cosine and tangent functions which are listed below.
sin (A + B) = sin A cos B + cosA sin B
sin (A ­ B) = sin A cos B ­ cosA sin B
cos (A + B) = cos A cos B ­ sin A sin B
cos (A ­ B) = cos A cos B + sin A sin B
Note: You do not need to memorize these identities. They will be given to you for tests & quizzes.
Example: Use the correct formula to find the exact value of:
a) tan 150
b)
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Example: Write each of the following expressions as a single trig
expression. Then give the exact value of the expression.
a)
b)
c)
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Example: Simplify to a single trig function.
There are also other identities called "double angle formulas".
sin 2A = 2 sin A cos A
cos 2A = cos2A ­ sin2A
cos 2A = 2 cos2A ­ 1
cos 2A = 1 ­ 2 sin2A
Note: Again, you do not need to memorize these identities. They will be given to you for tests & quizzes. But you must know how to use them!
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Example: Simplify each of the following expressions.
a) b)
Example: Simplify to a single primary trig function.
a) b)
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Example: If A and B are both in Quadrant I, and and
evaluate the following:
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Pages 306 ­ 307
# 1, 2, 4, 5, 7, 8(a,c,d), 9 (a, b(i)), 10, 15, 18, and 20(a)
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