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Section exercises

Verbal

Explain the basis for the cofunction identities and when they apply.

The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures x , the second angle measures π 2 x . Then sin x = cos ( π 2 x ) . The same holds for the other cofunction identities. The key is that the angles are complementary.

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Is there only one way to evaluate cos ( 5 π 4 ) ? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

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Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f ( x ) = sin ( x ) and g ( x ) = cos ( x ) . (Hint: 0 x = x )

sin ( x ) = sin x , so sin x is odd. cos ( x ) = cos ( 0 x ) = cos x , so cos x is even.

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Algebraic

For the following exercises, find the exact value.

sin ( 11 π 12 )

6 2 4

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tan ( 19 π 12 )

2 3

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For the following exercises, rewrite in terms of sin x and cos x .

sin ( x 3 π 4 )

2 2 sin x 2 2 cos x

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cos ( x + 2 π 3 )

1 2 cos x 3 2 sin x

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For the following exercises, simplify the given expression.

sec ( π 2 θ )

csc θ

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tan ( π 2 x )

cot x

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sin ( 2 x ) cos ( 5 x ) sin ( 5 x ) cos ( 2 x )

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tan ( 3 2 x ) tan ( 7 5 x ) 1 + tan ( 3 2 x ) tan ( 7 5 x )

tan ( x 10 )

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For the following exercises, find the requested information.

Given that sin a = 2 3 and cos b = 1 4 , with a and b both in the interval [ π 2 , π ) , find sin ( a + b ) and cos ( a b ) .

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Given that sin a = 4 5 , and cos b = 1 3 , with a and b both in the interval [ 0 , π 2 ) , find sin ( a b ) and cos ( a + b ) .

sin ( a b ) = ( 4 5 ) ( 1 3 ) ( 3 5 ) ( 2 2 3 ) = 4 6 2 15 cos ( a + b ) = ( 3 5 ) ( 1 3 ) ( 4 5 ) ( 2 2 3 ) = 3 8 2 15

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For the following exercises, find the exact value of each expression.

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

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cos ( cos 1 ( 2 2 ) + sin 1 ( 3 2 ) )

2 6 4

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tan ( sin 1 ( 1 2 ) cos 1 ( 1 2 ) )

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Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

cos ( π 2 x )

sin x

Graph of y=sin(x) from -2pi to 2pi.
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tan ( π 3 + x )

cot ( π 6 x )

Graph of y=cot(pi/6 - x) from -2pi to pi - in comparison to the usual y=cot(x) graph, this one is reflected across the x-axis and shifted by pi/6.
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tan ( π 4 x )

cot ( π 4 + x )

Graph of y=cot(pi/4 + x) - in comparison to the usual y=cot(x) graph, this one is shifted by pi/4.
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sin ( π 4 + x )

sin x 2 + cos x 2

Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.
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For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2 x = x + x . )

f ( x ) = sin ( 4 x ) sin ( 3 x ) cos x , g ( x ) = sin x cos ( 3 x )

They are the same.

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f ( x ) = cos ( 4 x ) + sin x sin ( 3 x ) , g ( x ) = cos x cos ( 3 x )

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f ( x ) = sin ( 3 x ) cos ( 6 x ) , g ( x ) = sin ( 3 x ) cos ( 6 x )

They are the different, try g ( x ) = sin ( 9 x ) cos ( 3 x ) sin ( 6 x ) .

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f ( x ) = sin ( 4 x ) , g ( x ) = sin ( 5 x ) cos x cos ( 5 x ) sin x

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f ( x ) = sin ( 2 x ) , g ( x ) = 2 sin x cos x

They are the same.

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f ( θ ) = cos ( 2 θ ) , g ( θ ) = cos 2 θ sin 2 θ

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f ( θ ) = tan ( 2 θ ) , g ( θ ) = tan θ 1 + tan 2 θ

They are the different, try g ( θ ) = 2 tan θ 1 tan 2 θ .

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f ( x ) = sin ( 3 x ) sin x , g ( x ) = sin 2 ( 2 x ) cos 2 x cos 2 ( 2 x ) sin 2 x

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f ( x ) = tan ( x ) , g ( x ) = tan x tan ( 2 x ) 1 tan x tan ( 2 x )

They are different, try g ( x ) = tan x tan ( 2 x ) 1 + tan x tan ( 2 x ) .

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Technology

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

sin ( 195° )

3 1 2 2 , or  0.2588

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cos ( 345° )

1 + 3 2 2 , or 0.9659

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Extensions

For the following exercises, prove the identities provided.

tan ( x + π 4 ) = tan x + 1 1 tan x

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

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

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

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

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

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cos ( x + h ) cos x h = cos x cos h 1 h sin x sin h h

cos ( x + h ) cos x h = cos x cosh sin x sinh cos x h = cos x ( cosh 1 ) sin x sinh h = cos x cos h 1 h sin x sin h h

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For the following exercises, prove or disprove the statements.

tan ( u + v ) = tan u + tan v 1 tan u tan v

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tan ( u v ) = tan u tan v 1 + tan u tan v

True

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tan ( x + y ) 1 + tan x tan x = tan x + tan y 1 tan 2 x tan 2 y

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If α , β , and γ are angles in the same triangle, then prove or disprove sin ( α + β ) = sin γ .

True. Note that sin ( α + β ) = sin ( π γ ) and expand the right hand side.

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If α , β , and y are angles in the same triangle, then prove or disprove tan α + tan β + tan γ = tan α tan β tan γ

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Questions & Answers

how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
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
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu

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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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