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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

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
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
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Solve 2cos x + 3sin x = 0.5
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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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