Sum and Difference Identities

In this section, you will:

Photo of Mt. McKinley.

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.

Using the Sum and Difference Formulas for Cosine

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, which we can review in the unit circle shown in [link].

Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.

We will begin with the sum and difference formulas for cosine, 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 )=cosαcosβsinαsinβ
   
Difference formula for cosine cos( αβ )=cosαcosβ+sinαsinβ

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 ( cosα,sinα )

and point Q

is at an angle of β

from the positive x-axis with coordinates ( cosβ,sinβ ).

Note the measure of angle POQ

is αβ.

Label two more points: A

at an angle of ( αβ )

from the positive x-axis with coordinates ( cos( αβ ),sin( αβ ) );

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.

Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.

We can find the distance from P

to Q

using the distance formula.* * *

d PQ = (cosαcosβ) 2 + (sinαsinβ) 2 = cos 2 α2cosαcosβ+ cos 2 β+ sin 2 α2sinαsinβ+ sin 2 β

Then we apply the Pythagorean identity and simplify.

= ( cos 2 α+ sin 2 α )+( cos 2 β+ sin 2 β )2cosαcosβ2sinαsinβ = 1+12cosαcosβ2sinαsinβ = 22cosαcosβ2sinαsinβ

Similarly, using the distance formula we can find the distance from A

to B.

d AB = (cos(αβ)1) 2 + (sin(αβ)0) 2 = cos 2 (αβ)2cos(αβ)+1+ sin 2 (αβ)

Applying the Pythagorean identity and simplifying we get:

= ( cos 2 (αβ)+ sin 2 (αβ) )2cos(αβ)+1 = 12cos(αβ)+1 = 22cos(αβ)

Because the two distances are the same, we set them equal to each other and simplify.

22cosαcosβ2sinαsinβ = 22cos(αβ) 22cosαcosβ2sinαsinβ = 22cos(αβ)

Finally we subtract 2

from both sides and divide both sides by −2.

cosαcosβ+sinαsinβ=cos(αβ)  

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

Sum and Difference Formulas for Cosine

These formulas can be used to calculate the cosine of sums and differences of angles.

cos(α+β)=cosαcosβsinαsinβ
cos(αβ)=cosαcosβ+sinαsinβ

Given two angles, find the cosine of the difference between the angles.

  1. Write the difference formula for cosine.
  2. Substitute the values of the given angles into the formula.
  3. Simplify.
Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

Using the formula for the cosine of the difference of two angles, find the exact value of cos( 5π 4 π 6 ).

Begin by writing the formula for the cosine of the difference of two angles. Then substitute the given values.

cos(αβ) = cosαcosβ+sinαsinβ cos( 5π 4 π 6 ) = cos( 5π 4 )cos( π 6 )+sin( 5π 4 )sin( π 6 ) = ( 2 2 )( 3 2 )( 2 2 )( 1 2 ) = 6 4 2 4 = 6 2 4

Keep in mind that we can always check the answer using a graphing calculator in radian mode.

Find the exact value of cos( π 3 π 4 ).

2 + 6 4
Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine

Find the exact value of cos(75°).

As 75°=45°+30°,

we can evaluate cos(75°)

as cos(45°+30°).

cos(α+β) = cosαcosβsinαsinβ cos(45°+30°) = cos(45°)cos(30°)sin(45°)sin(30°) = 2 2 ( 3 2 ) 2 2 ( 1 2 ) = 6 4 2 4 = 6 2 4

Keep in mind that we can always check the answer using a graphing calculator in degree mode.

Analysis

Note that we could have also solved this problem using the fact that 75°=135°60°.

cos(αβ) = cosαcosβ+sinαsinβ cos(135°60°) = cos(135°)cos(60°)+sin(135°)sin(60°) = ( 2 2 )( 1 2 )+( 2 2 )( 3 2 ) = ( 2 4 )+( 6 4 ) = ( 6 2 4 )

Find the exact value of cos(105°).

2 6 4

Using the Sum and Difference Formulas for Sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

Sum and Difference Formulas for Sine

These formulas can be used to calculate the sines of sums and differences of angles.

sin( α+β )=sinαcosβ+cosαsinβ
sin( αβ )=sinαcosβcosαsinβ

Given two angles, find the sine of the difference between the angles.

  1. Write the difference formula for sine.
  2. Substitute the given angles into the formula.
  3. Simplify.
Using Sum and Difference Identities to Evaluate the Difference of Angles

Use the sum and difference identities to evaluate the difference of the angles and show that part a equals part b.

  1. sin(45°30°)
  2. sin(135°120°)
  1. Let’s begin by writing the formula and substitute the given angles.
    sin(αβ) = sinαcosβcosαsinβ sin(45°30°) = sin(45°)cos(30°)cos(45°)sin(30°)

    Next, we need to find the values of the trigonometric expressions.

    sin(45°)= 2 2 , cos(30°)= 3 2 , cos(45°)= 2 2 , sin(30°)= 1 2

    Now we can substitute these values into the equation and simplify.

    sin(45°30°) = 2 2 ( 3 2 ) 2 2 ( 1 2 ) = 6 2 4
  2. Again, we write the formula and substitute the given angles.
    sin(αβ) = sinαcosβcosαsinβ 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 2

    Now 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
Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

Find the exact value of sin( cos −1 1 2 + sin −1 3 5 ).

Then check the answer with a graphing calculator.

The pattern displayed in this problem is sin( α+β ).

Let α= cos −1 1 2

and β= sin −1 3 5 .

Then we can write

cosα = 1 2 ,0απ sinβ = 3 5 , π 2 β π 2

We will use the Pythagorean identities to find sinα

and cosβ.

sinα = 1 cos 2 α = 1 1 4 = 3 4 = 3 2 cosβ = 1 sin 2 β = 1 9 25 = 16 25 = 4 5

Using the sum formula for sine,

sin( cos 1 1 2 + sin 1 3 5 ) = sin(α+β) = sinαcosβ+cosαsinβ = 3 2 4 5 + 1 2 3 5 = 4 3 +3 10

Using the Sum and Difference Formulas for Tangent

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, tanx= sinx cosx ,cosx0.

Let’s derive the sum formula for tangent.

tan(α+β) = sin(α+β) cos(α+β) = sinαcosβ+cosαsinβ cosαcosβsinαsinβ = sinαcosβ+cosαsinβ cosαcosβ cosαcosβsinαsinβ cosαcosβ Divide the numerator and denominator by cosαcosβ. = sinα cosβ cosα cosβ + cosα sinβ cosα cosβ cosα cosβ cosα cosβ sinαsinβ cosαcosβ = sinα cosα + sinβ cosβ 1 sinαsinβ cosαcosβ = tanα+tanβ 1tanαtanβ

We can derive the difference formula for tangent in a similar way.

Sum and Difference Formulas for Tangent

The sum and difference formulas for tangent are:

tan( α+β )= tanα+tanβ 1tanαtanβ
tan( αβ )= tanαtanβ 1+tanαtanβ

Given two angles, find the tangent of the sum of the angles.

  1. Write the sum formula for tangent.
  2. Substitute the given angles into the formula.
  3. Simplify.
Finding the Exact Value of an Expression Involving Tangent

Find the exact value of tan( π 6 + π 4 ).

Let’s first write the sum formula for tangent and then substitute the given angles into the formula.

tan(α+β) = tanα+tanβ 1tanαtanβ tan( π 6 + π 4 ) = tan( π 6 )+tan( π 4 ) 1( tan( π 6 ) )( tan( π 4 ) )

Next, we determine the individual function values within the formula:

tan( π 6 )= 1 3 ,tan( π 4 )=1

So we have

tan( π 6 + π 4 ) = 1 3 +1 1( 1 3 )(1) = 1+ 3 3 3 1 3 = 1+ 3 3 ( 3 3 1 ) = 3 +1 3 1

Find the exact value of tan( 2π 3 + π 4 ).

1 3 1+ 3
Finding Multiple Sums and Differences of Angles

Given  sinα= 3 5 ,0<α< π 2 ,cosβ= 5 13 ,π<β< 3π 2 ,

find

  1. sin( α+β )
  2. cos( α+β )
  3. tan( α+β )
  4. tan( αβ )

We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.

  1. To find sin( α+β ),

    we begin with

    sinα= 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

    Diagram of a triangle in the x,y plane. The vertices are at the origin, (4,0), and (4,3). The angle at the origin is alpha degrees, The angle formed by the x-axis and the side from (4,3) to (4,0) is a right angle. The side opposite the right angle has length 5.

    Since cosβ= 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 = ±12

    Since β

    is in the third quadrant, a=–12.

    Diagram of a triangle in the x,y plane. The vertices are at the origin, (-5,0), and (-5, -12). The angle at the origin is Beta degrees. The angle formed by the x axis and the side from (-5, -12) to (-5,0) is a right angle. The side opposite the right angle has length 13.

    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]: cosα= 4 5 .

    We can also find the sine of β

    from the triangle in [link], as opposite side over the hypotenuse: sinβ= 12 13 .

    Now we are ready to evaluate sin( α+β ).

    sin(α+β) = sinαcosβ+cosαsinβ = ( 3 5 )( 5 13 )+( 4 5 )( 12 13 ) = 15 65 48 65 = 63 65
  2. We can find cos( α+β )

    in a similar manner. We substitute the values according to the formula.

    cos(α+β) = cosαcosβsinαsinβ = ( 4 5 )( 5 13 )( 3 5 )( 12 13 ) = 20 65 + 36 65 = 16 65
  3. For tan( α+β ),

    if

    sinα= 3 5

    and

    cosα= 4 5 ,

    then

    tanα= 3 5 4 5 = 3 4

    If sinβ= 12 13

    and cosβ= 5 13 ,

    then

    tanβ= 12 13 5 13 = 12 5

    Then,

    tan(α+β) = tanα+tanβ 1tanαtanβ = 3 4 + 12 5 1 3 4 ( 12 5 ) =    63 20 16 20 = 63 16
  4. To find tan( αβ ),

    we have the values we need. We can substitute them in and evaluate.

    tan(αβ) = tanαtanβ 1+tanαtanβ = 3 4 12 5 1+ 3 4 ( 12 5 ) = 33 20 56 20 = 33 56
Analysis

A common mistake when addressing problems such as this one is that we may be tempted to think that α

and β

are angles in the same triangle, which of course, they are not. Also note that

tan( α+β )= sin( α+β ) cos( α+β )

Using Sum and Difference Formulas for Cofunctions

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 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 sinθ=cos( π 2 θ ),

which is opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of θ

equals the cofunction of the complement of θ.

Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.

Image of a right triangle. The remaining angles are labeled theta and pi/2 - theta.

From these relationships, the cofunction identities are formed. Recall that you first encountered these identities in The Unit Circle: Sine and Cosine Functions.

Cofunction Identities

The cofunction identities are summarized in [link].

sinθ=cos( π 2 θ ) cosθ=sin( π 2 θ )
tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ )
secθ=csc( π 2 θ ) cscθ=sec( π 2 θ )

Notice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using

cos( αβ )=cosαcosβ+sinαsinβ,

we can write

cos( π 2 θ ) = cos π 2 cosθ+sin π 2 sinθ = (0)cosθ+(1)sinθ = sinθ
Finding a Cofunction with the Same Value as the Given Expression

Write tan π 9

in terms of its cofunction.

The cofunction of tanθ=cot( π 2 θ ).

Thus,

tan( π 9 ) = cot( π 2 π 9 ) = cot( 9π 18 2π 18 ) = cot( 7π 18 )

Write sin π 7

in terms of its cofunction.

cos( 5π 14 )

Using the Sum and Difference Formulas to Verify Identities

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 presented earlier may help simplify the process of verifying an identity.

Given an identity, verify using sum and difference formulas.

  1. 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.
  2. Look for opportunities to use the sum and difference formulas.
  3. Rewrite sums or differences of quotients as single quotients.
  4. If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.
Verifying an Identity Involving Sine

Verify the identity sin(α+β)+sin(αβ)=2sinαcosβ.

We see that the left side of the equation includes the sines of the sum and the difference of angles.

sin(α+β) = sinαcosβ+cosαsinβ sin(αβ) = sinαcosβcosαsinβ

We can rewrite each using the sum and difference formulas.

sin(α+β)+sin(αβ) = sinαcosβ+cosαsinβ+sinαcosβcosαsinβ = 2sinαcosβ

We see that the identity is verified.

Verifying an Identity Involving Tangent

Verify the following identity.

sin(αβ) cosαcosβ =tanαtanβ

We can begin by rewriting the numerator on the left side of the equation.

sin(αβ) cosαcosβ = sinαcosβcosαsinβ cosαcosβ = sinα cosβ cosα cosβ cosα sinβ cosα cosβ Rewrite using a common denominator. = sinα cosα sinβ cosβ Cancel. = tanαtanβ Rewrite in terms of tangent.

We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.

Verify the identity: tan( πθ )=tanθ.

tan(πθ) = tan(π)tanθ 1+tan(π)tanθ = 0tanθ 1+0tanθ = tanθ
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

tanθ= 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.

tanθ = 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
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 tanβ= 47 50 ,

and tan( βα )= 40 50 = 4 5 .

We can then use difference formula for tangent.

tan( βα )= tanβtanα 1+tanβtanα

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

4 5 = 47 50 tanα 1+ 47 50 tanα 4( 1+ 47 50 tanα ) = 5( 47 50 tanα )

Use the distributive property, and then simplify the functions.

4(1)+4( 47 50 )tanα = 5( 47 50 )5tanα 4+3.76tanα = 4.75tanα 5tanα+3.76tanα = 0.7 8.76tanα = 0.7 tanα 0.07991 tan 1 (0.07991) .079741

Now we can calculate the angle in degrees.

α0.079741( 180 π )4.57°
Analysis

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.

Access these online resources for additional instruction and practice with sum and difference identities.

Key Equations

Sum Formula for Cosine cos( α+β )=cosαcosβsinαsinβ
   
Difference Formula for Cosine cos( αβ )=cosαcosβ+sinαsinβ
   
Sum Formula for Sine sin( α+β )=sinαcosβ+cosαsinβ
   
Difference Formula for Sine sin( αβ )=sinαcosβcosαsinβ
   
Sum Formula for Tangent tan( α+β )= tanα+tanβ 1tanαtanβ
   
Difference Formula for Tangent tan( αβ )= tanαtanβ 1+tanαtanβ
   
Cofunction identities sinθ = cos( π 2 θ ) cosθ = sin( π 2 θ ) tanθ = cot( π 2 θ ) cotθ = tan( π 2 θ ) secθ = csc( π 2 θ ) cscθ = sec( π 2 θ )

Key Concepts

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 sinx=cos( π 2 x ).

The same holds for the other cofunction identities. The key is that the angles are complementary.

Is there only one way to evaluate cos( 5π 4 )?

Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

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: 0x=x

)

sin( x )=sinx,

so sinx

is odd. cos( x )=cos( 0x )=cosx,

so cosx

is even.

Algebraic

For the following exercises, find the exact value.

cos( 7π 12 )
cos( π 12 )
2 + 6 4
sin( 5π 12 )
sin( 11π 12 )
6 2 4
tan( π 12 )
tan( 19π 12 )
2 3

For the following exercises, rewrite in terms of sinx

and cosx.

sin( x+ 11π 6 )
sin( x 3π 4 )
2 2 sinx 2 2 cosx
cos( x 5π 6 )
cos( x+ 2π 3 )
1 2 cosx 3 2 sinx

For the following exercises, simplify the given expression.

csc( π 2 t )
sec( π 2 θ )
cscθ
cot( π 2 x )
tan( π 2 x )
cotx
sin(2x)cos(5x)sin(5x)cos(2x)
tan( 3 2 x )tan( 7 5 x ) 1+tan( 3 2 x )tan( 7 5 x )
tan( x 10 )

For the following exercises, find the requested information.

Given that sina= 2 3

and cosb= 1 4 ,

with a

and b

both in the interval [ π 2 ,π ),

find sin(a+b)

and cos(ab).

Given that sina= 4 5 ,

and cosb= 1 3 ,

with a

and b

both in the interval [ 0, π 2 ),

find sin(ab)

and cos(a+b).

sin(ab) = ( 4 5 )( 1 3 )( 3 5 )( 2 2 3 ) = 46 2 15 cos(a+b) = ( 3 5 )( 1 3 )( 4 5 )( 2 2 3 ) = 38 2 15

For the following exercises, find the exact value of each expression.

sin( cos 1 (0) cos 1 ( 1 2 ) )
cos( cos 1 ( 2 2 )+ sin 1 ( 3 2 ) )
2 6 4
tan( sin 1 ( 1 2 ) cos 1 ( 1 2 ) )

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 )
sinx

Graph of y=sin(x) from -2pi to 2pi.

sin(πx)
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.

sin( π 3 +x )
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.

cos( 7π 6 +x )
sin( π 4 +x )
sinx 2 + cosx 2

Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.

cos( 5π 4 +x )

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.

)

f( x )=sin( 4x )sin( 3x )cosx,g( x )=sinxcos( 3x )

They are the same.

f( x )=cos( 4x )+sinxsin( 3x ),g( x )=cosxcos( 3x )
f( x )=sin( 3x )cos( 6x ),g( x )=sin( 3x )cos( 6x )

They are the different, try g( x )=sin( 9x )cos( 3x )sin( 6x ).

f(x)=sin(4x),g(x)=sin(5x)cosxcos(5x)sinx
f(x)=sin(2x),g(x)=2sinxcosx

They are the same.

f( θ )=cos( 2θ ),g( θ )= cos 2 θ sin 2 θ
f(θ)=tan(2θ),g(θ)= tanθ 1+ tan 2 θ

They are the different, try g( θ )= 2tanθ 1 tan 2 θ .

f(x)=sin(3x)sinx,g(x)= sin 2 (2x) cos 2 x cos 2 (2x) sin 2 x
f(x)=tan(x),g(x)= tanxtan(2x) 1tanxtan(2x)

They are different, try g( x )= tanxtan( 2x ) 1+tanxtan( 2x ) .

Technology

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

sin(75°)
sin(195°)
3 1 2 2 ,or 0.2588
cos(165°)
cos(345°)
1+ 3 2 2 ,

or 0.9659

tan(−15°)

Extensions

For the following exercises, prove the identities provided.

tan(x+ π 4 )= tanx+1 1tanx
tan( x+ π 4 ) = tanx+tan( π 4 ) 1tanxtan( π 4 ) = tanx+1 1tanx(1) = tanx+1 1tanx
tan(a+b) tan(ab) = sinacosa+sinbcosb sinacosasinbcosb
cos(a+b) cosacosb =1tanatanb
cos(a+b) cosacosb = cosacosb cosacosb sinasinb cosacosb = 1tanatanb
cos( x+y )cos( xy )= cos 2 x sin 2 y
cos(x+h)cosx h =cosx cosh1 h sinx sinh h
cos(x+h)cosx h = cosxcoshsinxsinhcosx h = cosx(cosh1)sinxsinh h = cosx cosh1 h sinx sinh h

For the following exercises, prove or disprove the statements.

tan(u+v)= tanu+tanv 1tanutanv
tan(uv)= tanutanv 1+tanutanv

True

tan( x+y ) 1+tanxtanx = tanx+tany 1 tan 2 x tan 2 y

If α, β,

and γ

are angles in the same triangle, then prove or disprove sin( α+β )=sinγ.

True. Note that sin( α+β )=sin( πγ )

and expand the right hand side.

If α,β,

and y

are angles in the same triangle, then prove or disprove tanα+tanβ+tanγ=tanαtanβtanγ


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