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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

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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