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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

0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply
tan20?×tan40?×tan80?
Santosh Reply
At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?
Kimberly Reply
-3 and -2
Julberte Reply
tan(?cosA)=cot(?sinA) then prove cos(A-?/4)=1/2?2
Chirag Reply
tan(pi.cosA)=cot(?sinA) then prove cos(A-?/4)=1/2?2
Chirag Reply
sin x(1+tan x)+cos x(1+cot x) = sec x +cosec
Ankit Reply
let p(x)xq
Sophie Reply
To the nearest whole number, what was the initial population in the culture?
Cheyenne Reply
do posible if one line is parallel
Fran Reply
The length is one inch more than the width, which is one inch more than the height. The volume is 268.125 cubic inches.
Vamprincess Reply
Using Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating ?T??T? and ?a.?
James Reply
cos(x-45)°=Sin x ;x=?
Samaresh Reply
10-n ft
Nalin 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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