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cos ( 6 t ) + cos ( 4 t )

2 cos ( 5 t ) cos t

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sin ( 3 x ) + sin ( 7 x )

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cos ( 7 x ) + cos ( 7 x )

2 cos ( 7 x )

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sin ( 3 x ) sin ( 3 x )

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cos ( 3 x ) + cos ( 9 x )

2 cos ( 6 x ) cos ( 3 x )

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sin h sin ( 3 h )

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For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

cos ( 45° ) cos ( 15° )

1 4 ( 1 + 3 )

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cos ( 45° ) sin ( 15° )

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sin ( −345° ) sin ( −15° )

1 4 ( 3 2 )

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sin ( 195° ) cos ( 15° )

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sin ( −45° ) sin ( −15° )

1 4 ( 3 1 )

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For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

cos ( 23° ) sin ( 17° )

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2 sin ( 100° ) sin ( 20° )

cos ( 80° ) cos ( 120° )

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2 sin ( −100° ) sin ( −20° )

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sin ( 213° ) cos ( )

1 2 ( sin ( 221° ) + sin ( 205° ) )

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2 cos ( 56° ) cos ( 47° )

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For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

sin ( 76° ) + sin ( 14° )

2 cos ( 31° )

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cos ( 58° ) cos ( 12° )

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sin ( 101° ) sin ( 32° )

2 cos ( 66.5° ) sin ( 34.5° )

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cos ( 100° ) + cos ( 200° )

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sin ( −1° ) + sin ( −2° )

2 sin ( −1.5° ) cos ( 0.5° )

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For the following exercises, prove the identity.

cos ( a + b ) cos ( a b ) = 1 tan a tan b 1 + tan a tan b

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4 sin ( 3 x ) cos ( 4 x ) = 2 sin ( 7 x ) 2 sin x

2 sin ( 7 x ) 2 sin x = 2 sin ( 4 x + 3 x ) 2 sin ( 4 x 3 x ) = 2 ( sin ( 4 x ) cos ( 3 x ) + sin ( 3 x ) cos ( 4 x ) ) 2 ( sin ( 4 x ) cos ( 3 x ) sin ( 3 x ) cos ( 4 x ) ) = 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) 2 sin ( 4 x ) cos ( 3 x ) + 2 sin ( 3 x ) cos ( 4 x ) ) = 4 sin ( 3 x ) cos ( 4 x )

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6 cos ( 8 x ) sin ( 2 x ) sin ( 6 x ) = −3 sin ( 10 x ) csc ( 6 x ) + 3

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

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

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

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

2 tan x cos ( 3 x ) = 2 sin x cos ( 3 x ) cos x = 2 ( .5 ( sin ( 4 x ) sin ( 2 x ) ) ) cos x = 1 cos x ( sin ( 4 x ) sin ( 2 x ) ) = sec x ( sin ( 4 x ) sin ( 2 x ) )

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cos ( a + b ) + cos ( a b ) = 2 cos a cos b

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Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

cos ( 58° ) + cos ( 12° )

2 cos ( 35° ) cos ( 23° ) , 1.5081

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sin ( ) sin ( )

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cos ( 44° ) cos ( 22° )

2 sin ( 33° ) sin ( 11° ) , 0.2078

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cos ( 176° ) sin ( )

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sin ( −14° ) sin ( 85° )

1 2 ( cos ( 99° ) cos ( 71° ) ) , −0.2410

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Technology

For the following exercises, algebraically determine whether each of the given equation is an identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

2 sin ( 2 x ) sin ( 3 x ) = cos x cos ( 5 x )

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cos ( 10 θ ) + cos ( 6 θ ) cos ( 6 θ ) cos ( 10 θ ) = cot ( 2 θ ) cot ( 8 θ )

It is an identity.

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sin ( 3 x ) sin ( 5 x ) cos ( 3 x ) + cos ( 5 x ) = tan x

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

It is not an identity, but 2 cos 3 x is.

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sin ( 2 x ) + sin ( 4 x ) sin ( 2 x ) sin ( 4 x ) = tan ( 3 x ) cot x

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For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

sin ( 9 t ) sin ( 3 t ) cos ( 9 t ) + cos ( 3 t )

tan ( 3 t )

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2 sin ( 8 x ) cos ( 6 x ) sin ( 2 x )

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sin ( 3 x ) sin x sin x

2 cos ( 2 x )

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cos ( 5 x ) + cos ( 3 x ) sin ( 5 x ) + sin ( 3 x )

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sin x cos ( 15 x ) cos x sin ( 15 x )

sin ( 14 x )

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Extensions

For the following exercises, prove the following sum-to-product formulas.

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

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cos x + cos y = 2 cos ( x + y 2 ) cos ( x y 2 )

Start with cos x + cos y . Make a substitution and let x = α + β and let y = α β , so cos x + cos y becomes cos ( α + β ) + cos ( α β ) = cos α cos β sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β

Since x = α + β and y = α β , we can solve for α and β in terms of x and y and substitute in for 2 cos α cos β and get 2 cos ( x + y 2 ) cos ( x y 2 ) .

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For the following exercises, prove the identity.

sin ( 6 x ) + sin ( 4 x ) sin ( 6 x ) sin ( 4 x ) = tan ( 5 x ) cot x

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cos ( 3 x ) + cos x cos ( 3 x ) cos x = cot ( 2 x ) cot x

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

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cos ( 6 y ) + cos ( 8 y ) sin ( 6 y ) sin ( 4 y ) = cot y cos ( 7 y ) sec ( 5 y )

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cos ( 2 y ) cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = tan y

cos ( 2 y ) cos ( 4 y ) sin ( 2 y ) + sin ( 4 y ) = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = 2 sin ( 3 y ) sin ( y ) 2 sin ( 3 y ) cos y = tan y

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sin ( 10 x ) sin ( 2 x ) cos ( 10 x ) + cos ( 2 x ) = tan ( 4 x )

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

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

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

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tan ( π 4 t ) = 1 tan t 1 + tan t

tan ( π 4 t ) = tan ( π 4 ) tan t 1 + tan ( π 4 ) tan ( t ) = 1 tan t 1 + tan t

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Questions & Answers

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply
Practice Key Terms 2

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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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