| title | layout |
|---|---|
Sum and Difference Identities |
page |
{: #Figure_07_02_001}
How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.
The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.
In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.
Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles{: data-type="term" .no-emphasis}, which we can review in the unit circle shown in [link].
{: #Figure_07_02_008}
We will begin with the sum and difference formulas for cosine{: data-type="term" .no-emphasis}, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See [link].
| 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
| {: #Table_07_02_01 summary="Two rows, two columns. The table has ordered pairs of these row values: (Sum formula for cosine, cos(a+B) = cos(a)cos(B) - sin(a)sin(B)) and (Difference formula for cosine, cos(a-B) = cos(a)cos(B) + sin(a)sin(B))."}
First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See [link]. Point P
is at an angle α
from the positive *x-*axis with coordinates ( cos α,sin α )
and point Q
is at an angle of β
from the positive *x-*axis with coordinates ( cos β,sin β ).
Note the measure of angle POQ
is α−β.
Label two more points: A
at an angle of ( α−β )
from the positive *x-*axis with coordinates ( cos( α−β ),sin( α−β ) );
and point B
with coordinates ( 1,0 ).
Triangle POQ
is a rotation of triangle AOB
and thus the distance from P
to Q
is the same as the distance from A
to B.
{: #Figure_07_02_002}
We can find the distance from P
to Q
using the distance formula{: data-type="term" .no-emphasis}.* * * {: data-type="newline"}
Then we apply the Pythagorean identity{: data-type="term" .no-emphasis} and simplify.
Similarly, using the distance formula we can find the distance from A
to B.
Applying the Pythagorean identity and simplifying we get:
Because the two distances are the same, we set them equal to each other and simplify.
Finally we subtract 2
from both sides and divide both sides by −2.
Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.
- Write the difference formula for cosine.
- Substitute the values of the given angles into the formula.
- Simplify. {: type="1"}
we can evaluate cos( 75 ∘ )
as cos( 45 ∘ + 30 ∘ ).
Thus,
The sum and difference formulas for sine{: data-type="term" .no-emphasis} can be derived in the same manner as those for cosine, and they resemble the cosine formulas.
- Write the difference formula for sine.
- Substitute the given angles into the formula.
- Simplify. {: type="1"}
-
sin( 45 ∘ − 30 ∘ )
-
sin( 135 ∘ − 120 ∘ ) {: type="a"}
Next, we need to find the values of the trigonometric expressions.
<div data-type="equation" class="unnumbered" data-label="">
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>sin</mi><mrow><mo>(</mo> <mrow> <msup> <mrow> <mn>45</mn> </mrow> <mo>∘</mo> </msup> </mrow> <mo>)</mo></mrow><mo>=</mo><mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>,</mo><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo> <mrow> <msup> <mrow> <mn>30</mn> </mrow> <mo>∘</mo> </msup> </mrow> <mo>)</mo></mrow><mo>=</mo><mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>,</mo><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo> <mrow> <msup> <mrow> <mn>45</mn> </mrow> <mo>∘</mo> </msup> </mrow> <mo>)</mo></mrow><mo>=</mo><mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>,</mo><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo> <mrow> <msup> <mrow> <mn>30</mn> </mrow> <mo>∘</mo> </msup> </mrow> <mo>)</mo></mrow><mo>=</mo><mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math>
</div>
Now we can substitute these values into the equation and simplify.
<div data-type="equation" class="unnumbered" data-label="">
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtable columnalign="left"> <mtr columnalign="left"> <mtd columnalign="left"> <mrow /> </mtd> </mtr> <mtr columnalign="left"> <mtd columnalign="left"> <mrow> <mi>sin</mi><mo stretchy="false">(</mo><msup> <mrow> <mn>45</mn> </mrow> <mo>∘</mo> </msup> <mo>−</mo><msup> <mrow> <mn>30</mn> </mrow> <mo>∘</mo> </msup> <mo stretchy="false">)</mo><mo>=</mo><mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mrow><mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo></mrow><mo>−</mo><mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mrow><mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo></mrow> </mrow> </mtd> </mtr> <mtr columnalign="left"> <mtd columnalign="left"> <mrow> <mtext> </mtext><mo>=</mo><mfrac> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>−</mo><msqrt> <mn>2</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </math>
</div>
-
Again, we write the formula and substitute the given angles.
sin(α−β)=sin α cos β−cos α sin β sin( 135 ∘ − 120 ∘ )=sin( 135 ∘ )cos( 120 ∘ )−cos( 135 ∘ )sin( 120 ∘ )Next, we find the values of the trigonometric expressions.
sin( 135 ∘ )= 2 2 ,cos( 120 ∘ )=− 1 2 ,cos( 135 ∘ )=− 2 2 ,sin( 120 ∘ )= 3 2Now we can substitute these values into the equation and simplify.
sin( 135 ∘ − 120 ∘ )= 2 2 ( − 1 2 )−( − 2 2 )( 3 2 ) = − 2 + 6 4 = 6 − 2 4 sin( 135 ∘ − 120 ∘ )= 2 2 ( − 1 2 )−( − 2 2 )( 3 2 ) = − 2 + 6 4 = 6 − 2 4
{: type="a"}
Let α= cos −1 1 2
and β= sin −1 3 5 .
Then we can write
and cos β.
Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, tan x= sin x cos x ,cos x≠0.
Let’s derive the sum formula for tangent.
We can derive the difference formula for tangent in a similar way.
- Write the sum formula for tangent.
- Substitute the given angles into the formula.
- Simplify. {: type="1"}
find
-
sin( α+β )
-
cos( α+β )
-
tan( α+β )
-
tan( α−β ) {: type="a"}
-
To find sin( α+β ),
we begin with sin α= 3 5
and 0<α< π 2 .
The side opposite α
has length 3, the hypotenuse has length 5, and α
is in the first quadrant. See [link]. Using the Pythagorean Theorem, we can find the length of side a:
a 2 + 3 2 = 5 2 a 2 =16 a=4
{: #Figure_07_02_003}Since cos β=− 5 13
and π<β< 3π 2 ,
the side adjacent to β
is −5,
the hypotenuse is 13, and β
is in the third quadrant. See [link]. Again, using the Pythagorean Theorem, we have
( −5 ) 2 + a 2 = 13 2 25+ a 2 =169 a 2 =144 a=±12Since β
is in the third quadrant, a=–12.
{: #Figure_07_02_004}The next step is finding the cosine of α
and the sine of β.
The cosine of α
is the adjacent side over the hypotenuse. We can find it from the triangle in [link]: cos α= 4 5 .
We can also find the sine of β
from the triangle in [link], as opposite side over the hypotenuse: sin β=− 12 13 .
Now we are ready to evaluate sin( α+β ).
sin(α+β)=sin αcos β+cos αsin β =( 3 5 )( − 5 13 )+( 4 5 )( − 12 13 ) =− 15 65 − 48 65 =− 63 65 -
We can find cos( α+β )
in a similar manner. We substitute the values according to the formula.
cos(α+β)=cos α cos β−sin α sin β =( 4 5 )( − 5 13 )−( 3 5 )( − 12 13 ) =− 20 65 + 36 65 = 16 65 -
For tan( α+β ),
if sin α= 3 5
and cos α= 4 5 ,
then
tan α= 3 5 4 5 = 3 4If sin β=− 12 13
and cos β=− 5 13 ,
then
tan β= −12 13 −5 13 = 12 5Then,
tan(α+β)= tan α+tan β 1−tan α tan β = 3 4 + 12 5 1− 3 4 ( 12 5 ) = 63 20 − 16 20 =− 63 16 -
To find tan( α−β ),
we have the values we need. We can substitute them in and evaluate.
tan( α−β )= tan α−tan β 1+tan α tan β = 3 4 − 12 5 1+ 3 4 ( 12 5 ) = − 33 20 56 20 =− 33 56
{: type="a"}
and β
are angles in the same triangle, which of course, they are not. Also note that
Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry{: .target-chapter} that, if the sum of two positive angles is π 2 ,
those two angles are complements, and the sum of the two acute angles in a right triangle is π 2 ,
so they are also complements. In [link], notice that if one of the acute angles is labeled as θ,
then the other acute angle must be labeled ( π 2 −θ ).
Notice also that sin θ=cos( π 2 −θ ):
opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of θ
equals the cofunction{: data-type="term" .no-emphasis} of the complement of θ.
Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.
{: #Figure_07_02_007}
From these relationships, the cofunction identities{: data-type="term" .no-emphasis} are formed.
| 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 ) |
Notice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using
we can write
in terms of its cofunction.
Thus,
in terms of its cofunction.
Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules from Solving Trigonometric Equations with Identities{: .target-chapter} may help simplify the process of verifying an identity.
- Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.
- Look for opportunities to use the sum and difference formulas.
- Rewrite sums or differences of quotients as single quotients.
- If the process becomes cumbersome, rewrite the expression in terms of sines and cosines. {: type="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
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 .
)
{: #Figure_07_02_005}
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].
{: #Figure_07_02_006}
and tan( β−α )= 40 50 = 4 5 .
We can then use difference formula for tangent.
Occasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.
<tr>
<td>Difference Formula for Cosine</td>
<td><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>cos</mi><mrow><mo>(</mo>
<mrow>
<mi>α</mi><mo>−</mo><mi>β</mi>
</mrow>
<mo>)</mo></mrow><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi>
</mrow>
</math>
</td>
</tr>
<tr>
<td>Sum Formula for Sine</td>
<td><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>sin</mi><mrow><mo>(</mo>
<mrow>
<mi>α</mi><mo>+</mo><mi>β</mi>
</mrow>
<mo>)</mo></mrow><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>+</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi>
</mrow>
</math>
</td>
</tr>
<tr>
<td>Difference Formula for Sine</td>
<td><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>sin</mi><mrow><mo>(</mo>
<mrow>
<mi>α</mi><mo>−</mo><mi>β</mi>
</mrow>
<mo>)</mo></mrow><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>β</mi><mo>−</mo><mi>cos</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>β</mi>
</mrow>
</math>
</td>
</tr>
<tr>
<td>Sum Formula for Tangent</td>
<td><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>tan</mi><mrow><mo>(</mo>
<mrow>
<mi>α</mi><mo>+</mo><mi>β</mi>
</mrow>
<mo>)</mo></mrow><mo>=</mo><mfrac>
<mrow>
<mi>tan</mi><mtext> </mtext><mi>α</mi><mo>+</mo><mi>tan</mi><mtext> </mtext><mi>β</mi>
</mrow>
<mrow>
<mn>1</mn><mo>−</mo><mi>tan</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>β</mi>
</mrow>
</mfrac>
</mrow>
</math>
</td>
</tr>
<tr>
<td>Difference Formula for Tangent</td>
<td><math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>tan</mi><mrow><mo>(</mo>
<mrow>
<mi>α</mi><mo>−</mo><mi>β</mi>
</mrow>
<mo>)</mo></mrow><mo>=</mo><mfrac>
<mrow>
<mi>tan</mi><mtext> </mtext><mi>α</mi><mo>−</mo><mi>tan</mi><mtext> </mtext><mi>β</mi>
</mrow>
<mrow>
<mn>1</mn><mo>+</mo><mi>tan</mi><mtext> </mtext><mi>α</mi><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>β</mi>
</mrow>
</mfrac>
</mrow>
</math>
</td>
</tr>
<tr>
<td>Cofunction identities</td>
<td><math xmlns="http://www.w3.org/1998/Math/MathML">
<mtable columnalign="left">
<mtr>
<mtd>
<mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo><mi>θ</mi>
</mrow>
<mo>)</mo></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>sin</mi><mrow><mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo><mi>θ</mi>
</mrow>
<mo>)</mo></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>cot</mi><mrow><mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo><mi>θ</mi>
</mrow>
<mo>)</mo></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>cot</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>tan</mi><mrow><mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo><mi>θ</mi>
</mrow>
<mo>)</mo></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>sec</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo><mi>θ</mi>
</mrow>
<mo>)</mo></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>csc</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mi>sec</mi><mrow><mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo><mi>θ</mi>
</mrow>
<mo>)</mo></mrow>
</mtd>
</mtr>
</mtable>
</math>
</td>
</tr>
</tbody></table>
- 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].
the second angle measures π 2 −x.
Then sinx=cos( π 2 −x ).
The same holds for the other cofunction identities. The key is that the angles are complementary.
Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.
and g(x)=cos(x).
(Hint: 0−x=−x
)
so sinx
is odd. cos( −x )=cos( 0−x )=cosx,
so cosx
is even.
For the following exercises, find the exact value.
For the following exercises, rewrite in terms of sin x
and cos x.
For the following exercises, simplify the given expression.
For the following exercises, find the requested information.
and cos b=− 1 4 ,
with a
and b
both in the interval [ π 2 ,π ),
find sin(a+b)
and cos(a−b).
and cos b= 1 3 ,
with a
and b
both in the interval [ 0, π 2 ),
find sin(a−b)
and cos(a+b).
{: data-type="newline"}
cos(a+b)=( 3 5 )( 1 3 )−( 4 5 )( 2 2 3 )= 3−8 2 15
For the following exercises, find the exact value of each expression.
For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.
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 2x=x+x.
)
For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.
or 0.9659
For the following exercises, prove the identities provided.
For the following exercises, prove or disprove the statements.
and γ
are angles in the same triangle, then prove or disprove sin( α+β )=sin γ.
and expand the right hand side.
and y
are angles in the same triangle, then prove or disprove tan α+tan β+tan γ=tan α tan β tan γ
| Sum Formula for Cosine | cos( α+β )=cos α cos β−sin αsin β |