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Verify the identity: tan ( π θ ) = tan θ .

tan ( π θ ) = tan ( π ) tan θ 1 + tan ( π ) tan θ = 0 tan θ 1 + 0 tan θ = tan θ
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Using sum and difference formulas to solve an application problem

Let L 1 and L 2 denote two non-vertical intersecting lines, and let θ denote the acute angle between L 1 and L 2 . See [link] . Show that

tan θ = m 2 m 1 1 + m 1 m 2

where m 1 and m 2 are the slopes of L 1 and L 2 respectively. ( Hint: Use the fact that tan θ 1 = m 1 and tan θ 2 = m 2 . )

Diagram of two non-vertical intersecting lines L1 and L2 also intersecting the x-axis. The acute angle formed by the intersection of L1 and L2 is theta. The acute angle formed by L2 and the x-axis is theta 1, and the acute angle formed by the x-axis and L1 is theta 2.

Using the difference formula for tangent, this problem does not seem as daunting as it might.

tan θ = tan ( θ 2 θ 1 ) = tan θ 2 tan θ 1 1 + tan θ 1 tan θ 2 = m 2 m 1 1 + m 1 m 2
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Investigating a guy-wire problem

For a climbing wall, a guy-wire R is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire S attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle α between the wires. See [link] .

Two right triangles. Both share the same base, 50 feet. The first has a height of 40 ft and hypotenuse S. The second has height 47 ft and hypotenuse R. The height sides of the triangles are overlapping. There is a B degree angle between R and the base, and an a degree angle between the two hypotenuses within the B degree angle.

Let’s first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are known, we can use the tangent function. Notice that tan β = 47 50 , and tan ( β α ) = 40 50 = 4 5 . We can then use difference formula for tangent.

tan ( β α ) = tan β tan α 1 + tan β tan α

Now, substituting the values we know into the formula, we have

4 5 = 47 50 tan α 1 + 47 50 tan α 4 ( 1 + 47 50 tan α ) = 5 ( 47 50 tan α )

Use the distributive property, and then simplify the functions.

4 ( 1 ) + 4 ( 47 50 ) tan α = 5 ( 47 50 ) 5 tan α 4 + 3.76 tan α = 4.7 5 tan α 5 tan α + 3.76 tan α = 0.7 8.76 tan α = 0.7 tan α 0.07991 tan 1 ( 0.07991 ) .079741

Now we can calculate the angle in degrees.

α 0.079741 ( 180 π ) 4.57°
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Access these online resources for additional instruction and practice with sum and difference identities.

Key equations

Sum Formula for Cosine cos ( α + β ) = cos α cos β sin α sin β
Difference Formula for Cosine cos ( α β ) = cos α cos β + sin α sin β
Sum Formula for Sine sin ( α + β ) = sin α cos β + cos α sin β
Difference Formula for Sine sin ( α β ) = sin α cos β cos α sin β
Sum Formula for Tangent tan ( α + β ) = tan α + tan β 1 tan α tan β
Difference Formula for Tangent tan ( α β ) = tan α tan β 1 + tan α tan β
Cofunction identities sin θ = cos ( π 2 θ ) cos θ = sin ( π 2 θ ) tan θ = cot ( π 2 θ ) cot θ = tan ( π 2 θ ) sec θ = csc ( π 2 θ ) csc θ = sec ( π 2 θ )

Key concepts

  • The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
  • The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. See [link] and [link] .
  • The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle. See [link] .
  • The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See [link] .
  • The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles. See [link] .
  • The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See [link] .
  • The cofunction identities apply to complementary angles and pairs of reciprocal functions. See [link] .
  • Sum and difference formulas are useful in verifying identities. See [link] and [link] .
  • Application problems are often easier to solve by using sum and difference formulas. See [link] and [link] .

Questions & Answers

(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
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Mark Reply
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Miranda
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
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Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
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but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
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Miranda
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Miranda
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Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
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Miranda
gotta practice, holmie
Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
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Ye
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Nokwanda
C'est comment
Abdel
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ANSHU
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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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