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csc 2 t = 3

sin 1 ( 3 3 ) , π sin 1 ( 3 3 ) , π + sin 1 ( 3 3 ) , 2 π sin 1 ( 3 3 )

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2 sin θ = −1

7 π 6 , 11 π 6

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tan x sin x + sin ( x ) = 0

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9 sin ω 2 = 4 sin 2 ω

sin 1 ( 1 4 ) , π sin 1 ( 1 4 )

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1 2 tan ( ω ) = tan 2 ( ω )

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For the following exercises, use basic identities to simplify the expression.

sec x cos x + cos x 1 sec x

1

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sin 3 x + cos 2 x sin x

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For the following exercises, determine if the given identities are equivalent.

sin 2 x + sec 2 x 1 = ( 1 cos 2 x ) ( 1 + cos 2 x ) cos 2 x

Yes

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tan 3 x csc 2 x cot 2 x cos x sin x = 1

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Sum and Difference Identities

For the following exercises, find the exact value.

tan ( 7 π 12 )

2 3

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sin ( 70° ) cos ( 25° ) cos ( 70° ) sin ( 25° )

2 2

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cos ( 83° ) cos ( 23° ) + sin ( 83° ) sin ( 23° )

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For the following exercises, prove the identity.

cos ( 4 x ) cos ( 3 x ) cos x = sin 2 x 4 cos 2 x sin 2 x

cos ( 4 x ) cos ( 3 x ) cos x = cos ( 2 x + 2 x ) cos ( x + 2 x ) cos x = cos ( 2 x ) cos ( 2 x ) sin ( 2 x ) sin ( 2 x ) cos x cos ( 2 x ) cos x + sin x sin ( 2 x ) cos x = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x ( cos 2 x sin 2 x ) + sin x ( 2 ) sin x cos x cos x = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x ( cos 2 x sin 2 x ) + 2 sin 2 x cos 2 x = cos 4 x 2 cos 2 x sin 2 x + sin 4 x 4 cos 2 x sin 2 x cos 4 x + cos 2 x sin 2 x + 2 sin 2 x cos 2 x = sin 4 x 4 cos 2 x sin 2 x + cos 2 x sin 2 x = sin 2 x ( sin 2 x + cos 2 x ) 4 cos 2 x sin 2 x = sin 2 x 4 cos 2 x sin 2 x

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cos ( 3 x ) cos 3 x = cos x sin 2 x sin x sin ( 2 x )

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For the following exercise, simplify the expression.

tan ( 1 2 x ) + tan ( 1 8 x ) 1 tan ( 1 8 x ) tan ( 1 2 x )

tan ( 5 8 x )

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For the following exercises, find the exact value.

cos ( sin 1 ( 0 ) cos 1 ( 1 2 ) )

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tan ( sin 1 ( 0 ) + sin 1 ( 1 2 ) )

3 3

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Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given cos θ = 1 3 and θ is in the interval [ π 2 , π ] .

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Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given sec θ = 5 3 and θ is in the interval [ π 2 , π ] .

24 25 , 7 25 , 24 7

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sec ( 3 π 8 )

2 ( 2 + 2 )

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For the following exercises, use [link] to find the desired quantities.

Image of a right triangle. The base is 24, the height is unknown, and the hypotenuse is 25. The angle opposite the base is labeled alpha, and the remaining acute angle is labeled beta.

sin ( 2 β ) , cos ( 2 β ) , tan ( 2 β ) , sin ( 2 α ) , cos ( 2 α ) , and  tan ( 2 α )

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sin ( β 2 ) , cos ( β 2 ) , tan ( β 2 ) , sin ( α 2 ) , cos ( α 2 ) , and  tan ( α 2 )

2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4

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For the following exercises, prove the identity.

2 cos ( 2 x ) sin ( 2 x ) = cot x tan x

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cot x cos ( 2 x ) = sin ( 2 x ) + cot x

cot x cos ( 2 x ) = cot x ( 1 2 sin 2 x ) = cot x cos x sin x ( 2 ) sin 2 x = 2 sin x cos x + cot x = sin ( 2 x ) + cot x

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For the following exercises, rewrite the expression with no powers.

cos 2 x sin 4 ( 2 x )

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tan 2 x sin 3 x

10 sin x 5 sin ( 3 x ) + sin ( 5 x ) 8 ( cos ( 2 x ) + 1 )

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Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

cos ( π 3 ) sin ( π 4 )

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2 sin ( 2 π 3 ) sin ( 5 π 6 )

3 2

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2 cos ( π 5 ) cos ( π 3 )

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For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

sin ( π 12 ) sin ( 7 π 12 )

2 2

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cos ( 5 π 12 ) + cos ( 7 π 12 )

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For the following exercises, change the functions from a product to a sum or a sum to a product.

sin ( 9 x ) cos ( 3 x )

1 2 ( sin ( 6 x ) + sin ( 12 x ) )

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cos ( 7 x ) cos ( 12 x )

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sin ( 11 x ) + sin ( 2 x )

2 sin ( 13 2 x ) cos ( 9 2 x )

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cos ( 6 x ) + cos ( 5 x )

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Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

tan x + 1 = 0

3 π 4 , 7 π 4

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2 sin ( 2 x ) + 2 = 0

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For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

2 sin 2 x sin x = 0

0 , π 6 , 5 π 6 , π

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cos 2 x cos x 1 = 0

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2 sin 2 x + 5 sin x + 3 = 0

3 π 2

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cos x 5 sin ( 2 x ) = 0

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1 sec 2 x + 2 + sin 2 x + 4 cos 2 x = 0

No solution

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Questions & Answers

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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