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Reciprocal Identities
Pythagorean Identities
Co-function Identities
Addition Formulas
sin (a + b) = sin a cos b + cos a sin b
cos (a + b) = cos a cos b sin a sin b
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Subtraction Formulas
sin (a - b) = sin a cos b - cos a sin b
cos (a - b) = cos a cos b + sin a sin b
Double Angle Formulas
sin 2a = 2 sin a cos a
cos 2a = cos 2a sin 2a = 2 cos 2a 1 = 1 2 sin 2a
Even-odd Identities
Sum-to-product Formulas
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Product-to-sum Formulas
12.b.
Take thecosine of to get
Take thecotangent of to get 1.
Take thecosecant of to get 2.
Multiply by 1 to get
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Multiply by 2 to get 1.
x=(1)
Remove the parentheses around the expression1.
x=1
12.c.
12.e.
Squaring an expression is the same as multiplying the expression by itself2 times.
Multiply each term in the first group by each term in the second group using the FOILmethod. FOIL stands for FirstOuterInnerLast, and is a method of multiplying two
binomials. First, multiply the first two terms in each binomial group. Next,multiply the
outer terms in each group, followed by the inner terms. Finally, multiply the last two
terms in each group.
Simplify the FOILexpression by multiplying and combining all like terms.
Squaring an expression is the same as multiplying the expression by itself2 times.
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Multiply each term in the first group by each term in the second group using the FOIL
method. FOIL stands for FirstOuterInnerLast, and is a method of multiplying two
binomials. First, multiply the first two terms in each binomial group. Next,multiply the
outer terms in each group, followed by the inner terms. Finally, multiply the last twoterms in each group.
Simplify the FOILexpression by multiplying and combining all like terms.
Remove the parentheses that are not needed from the expression.
Since and are like terms, add to to get
Since 4cos(x)sin(x) and -4cos(x)sin(x) are like terms, add -4cos(x)sin(x) to
4cos(x)sin(x) to get 0.
Since and are like terms, add to to get
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Replace with 5 using the identity
5=y
Since y is on the right-hand side of theequation, switch the sides so it is on the left-
hand side of the equation.
y=5
12.f.
Invert the trigonometric factorin the denominator.
Invert the trigonometric factorin the denominator.
Take thecosine of50 to get 0.64.
Expand the exponent(2) to theexpression.
Squaring a number is the same as multiplying the number by itself In
this case, 0.64 squared is 0.41.
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Remove the parentheses around the expression0.41.
Take thecosine of40 to get 0.77.
Expand the exponent(2) to theexpression.
Squaring a number is the same as multiplying the number by itself In
this case, 0.77 squared is 0.59.
0.41+(0.59)=y
Remove the parentheses around the expression0.59.
0.41+0.59=y
Add 0.59 to 0.41 to get 1.
1=y
Since y is on the right-hand side of theequation, switch the sides so it is on the left-hand side of the equation.
y=1
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12.g.
Take thecosine of110 to get -0.34.
Take thecosine of160 to get -0.94.
Multiply-1 by the -0.94 inside the parentheses.
Add 0.94 to -0.34 to get 0.6.
Take thesine of70 to get 0.94.
Take thecosine of20 to get 0.94.
Add 0.94 to 0.94 to get 1.88.
Expand the exponent(2) to theexpression.
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Squaring a number is the same as multiplying the number by itself In this
case, 0.6 squared is 0.36.
Expand the exponent(2) to theexpression.
Squaring a number is the same as multiplying the number by itself In
this case, 1.88 squared is 3.53.
(0.36)+(3.53)=y
Remove the parentheses that are not needed from the expression.
0.36+3.53=y
Add 3.53 to 0.36 to get 3.89.
3.89=y
Since y is on the right-hand side of theequation, switch the sides so it is on the left-
hand side of the equation.
y=3.89
12.h.
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Invert the trigonometric factorin the denominator.
To add fractions, the denominators must be equal. Thedenominators can be made equal
by finding the least common denominator(LCD). In this case, the LCD is
Next, multiply each fraction by a factorof1 that will create the LCD in each of the
fractions.
Complete the multiplication to produce a denominatorof in each expression.
Combine thenumerators of all expressions that have common denominators.
Take thecosine of75 to get 0.259.
Expand the exponent(2) to theexpression.
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Squaring a number is the same as multiplying the number by itself
In this case, 0.259 squared is 0.067.
Remove the parentheses around the expression0.067.
Take thetangent of75 to get 3.732.
Expand the exponent(2) to theexpression.
Squaring a number is the same as multiplying the number by itself
In this case, 3.732 squared is 13.928.
Remove the parentheses around the expression13.928.
Multiply0.067 by 13.928 to get 0.933.
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Multiply-1 by the 0.933 inside the parentheses.
Subtract 0.933 from 1 to get 0.067.
Take thecosine of75 to get 0.259.
Expand the exponent(2) to theexpression.
Squaring a number is the same as multiplying the number by itself
In this case, 0.259 squared is 0.067.
Divide0.067 by 0.067 to get 1.
1=x
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Since x is on the right-hand side of theequation, switch the sides so it is on the left-
hand side of the equation.
x=1
13.a.
Replace with an equivalentexpression using the fundamental
identities.
Rewrite the expression using the identity.
13.c.
Multiply by each term inside the parentheses.
Remove the parentheses around the expression
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Replace with an equivalentexpression using the fundamental identities.
Replace with an equivalentexpression using the fundamental
identities.
Remove the common factors that were cancelled out.
Combine all similarvariables in the expression.
To add fractions, the denominators must be equal. The denominators can be made
equal by finding the least common denominator(LCD). In this case, the LCD is
Next, multiply each fraction by a factorof1 that will create the LCD in
each of the fractions.
Complete the multiplication to produce a denominatorof in each
expression.
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Combine the numerators of all expressions that have common denominators.
Replace with 1 using the identity
Replace with an equivalentexpression in the numerator.
Since x is on the right-hand side of the equation, switch the sides so it is on the
left-hand side of the equation.
13.e.
Since 1 does not contain the variable to solve for, move it to the right-hand side
of the equation by subtracting 1 from both sides.
Simplify the right-hand side of the equation.
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Multiply each term in the equation by -1.
Multiply-x by -1 to get x.
Simplify the right-hand side of the equation by simplifying each term.
Replace with using the identity
Factorout the GCF of from each term in the polynomial.
Factorout the GCF of from
Reduce the expression by canceling out the common factorof from
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the numeratorand denominator.
Reduce the expression by canceling out the common factorof from
the numeratorand denominator.
13.f.
Replace with using the identity
Replace with using the identity
Remove the extra parentheses around the factors.
Multiply by each term inside the parentheses.
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Rewrite the factorof using the identity which in this case leads to
Reduce the expression by canceling out the common factors.
1=x
Since x is on the right-hand side of theequation, switch the sides so it is on the left-
hand side of the equation.
x=1
13.g.
Since x is on the right-hand side of the equation, switch the sides so it is on the
left-hand side of the equation.
Multiply each term by a factorof1 that will equate all the denominators. In this
case, all terms need a denominatorof The
expression needs to be multiplied by to make the denominator
The expression needs to be multiplied by
to make the denominator
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Multiply the expression by a factorof1 to create the least common denominator
(LCD) of
Remove the parentheses around the expression
Multiply the expression by a factorof1 to create the least common denominator
(LCD) of
Remove the parentheses around the expression
The numerators ofexpressions that have equal denominators can be combined. In
this case, and have the same
denominatorof so the numerators can be combined.
Simplify the numeratorof the expression.
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Since and are like terms, subtract from to get 0.
13.h.
Convert the to its equivalent.
Multiply by each term inside the parentheses.
Remove the parentheses around the expression
Since x is on the right-hand side of the equation, switch the sides so it is on the
left-hand side of the equation.
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14.a