Addition and Subtraction Formulas

Section 3.2 Addition and Subtraction Formulas

In Section 1.4, we used right triangles to determine the deviation of a waʻa (canoe) from its course based on the angle of deviation. If a waʻa sails for 120 nautical miles (NM), we were able to calculate the deviation from its course using right triangles to get the equation:

deviation = (120 NM) \cdot \sin (θ) .

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Before setting sail, a voyager studies a table listing the deviation distances corresponding to different houses of deviation. It’s crucial to understand that while adding angles may yield a third angle, adding their corresponding deviations will not accurately determine the total deviation. In other words:

\sin (α) + \sin (β) \neq \sin (α + β)

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for some angles of deviation

α

and

β .

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To illustrate this, let’s calculate the deviation distances for 1, 2, and 3 houses respectively. Using the given formula, we have:

\begin{aligned} 120 \sin (1 house) & = 120 \sin ({11.25}^{\circ}) \approx 23.4 NM \\ 120 \sin (2 houses) & = 120 \sin ({22.5}^{\circ}) \approx 45.9 NM \\ 120 \sin (3 houses) & = 120 \sin ({33.75}^{\circ}) \approx 66.7 NM \end{aligned}

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However,

120 \sin (1 house) + 120 \sin (2 houses) \approx 23.4 + 45.9 NM \approx 69.3 NM,

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which differs from the actual deviation of 66.7 NM when deviating by 3 houses.

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hese calculations demonstrate that the deviation distances for multiple houses cannot be determined by simply adding individual deviations, highlighting the importance of understanding trigonometric principles for accurate navigation. In this section, we will explore formulas for the addition and subtraction of angles in trigonometric functions.

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Subsection 3.2.1 Addition and Subtraction Formulas for Cosine

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First we will derive the addition and subtraction formulas for the cosine function.

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Definition 3.2.1. Addition and Subtraction Formulas for Cosine.

\begin{aligned} \cos (α + β) & = \cos α \cos β - \sin α \sin β \\ \cos (α - β) & = \cos α \cos β + \sin α \sin β \end{aligned}

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Proof.

First we will prove the Subtraction Formula for Cosine

\cos (α - β) = \cos α \cos β + \sin α \sin β .

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We begin by considering two points on the unit circle. Point

P

is at an angle of

β

in standard position with coordinates

(\cos β, \sin β)

and point

Q

is at an angle of

α

in standard position with coordinates

(\cos α, \sin α) .

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We use the Distance Formula to calculate the distance between

P

and

Q

to get

\begin{aligned} d (P, Q) & = \sqrt{(\cos α - \cos β)^{2} + (\sin α - \sin β)^{2}} \\ = \sqrt{\cos^{2} α - 2 \cos α \cos β + \cos^{2} β + \sin^{2} α - 2 \sin α \sin β + \sin^{2} β} \\ = \sqrt{(\cos^{2} α + \sin^{2} α) + (\cos^{2} β + \sin^{2} β) - 2 \cos α \cos β - 2 \sin α \sin β} \end{aligned}

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Then by the Pythagorean Identity (Definition 1.5.20),

\cos^{2} α + \sin^{2} α = 1

and

\cos^{2} β + \sin^{2} β = 1 .

Thus the distance becomes

\begin{aligned} d (P, Q) & = \sqrt{1 + 1 - 2 \cos α \cos β - 2 \sin α \sin β} \\ = \sqrt{2 - 2 \cos α \cos β - 2 \sin α \sin β} \end{aligned}

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Next, consider two additional points on a second unit circle. Point

A

has coordinates at

(1, 0)

and Point

B

is at an angle of

α - β

in standard position with coordinates

(\cos (α - β), \sin (α - β)) .

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The distance between

A

and

B

\begin{aligned} d (A, B) & = \sqrt{(\cos (α - β) - 1)^{2} + (\sin (α - β) - 0)^{2}} \\ = \sqrt{\cos^{2} (α - β) - 2 \cos (α - β) + 1 + \sin^{2} (α - β)} \\ = \sqrt{(\cos^{2} (α - β) + \sin^{2} (α - β)) - 2 \cos (α - β) + 1} \\ = \sqrt{1 - 2 \cos (α - β) + 1} \\ = \sqrt{2 - 2 \cos (α - β)} \end{aligned}

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Note that since

O P,

O Q,

O A,

and

O B

are lines from the center to points on the unit circle, they are congruent and have length of 1. Also note that

∠ P O B = ∠ A O B = α - β .

Then since two sides and the included angle of

Δ O P Q

and

Δ O A B

are equivalent, we can conclude by the Side-Angle-Side Theorem (SAS) in geometry that the two triangles are congruent. Thus, corresponding sides have the same lengths, giving us:

d (P Q) = d (A B) .

Substituting our results for

d (P, Q)

and

d (A, B)

we get

\begin{aligned} d (P Q) & = d (A B) \\ \sqrt{2 - 2 \cos α \cos β - 2 \sin α \sin β} & = \sqrt{2 - 2 \cos (α - β)} \\ 2 - 2 \cos α \cos β - 2 \sin α \sin β & = 2 - 2 \cos (α - β) \\ - 2 \cos α \cos β - 2 \sin α \sin β & = - 2 \cos (α - β) \end{aligned}

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Dividing both sides by -2 we arrive at the Subtraction Formula for Cosine

\cos α \cos β + \sin α \sin β = \cos (α - β)

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To prove the Addition Formula for Cosine, replace

β

with

- β

in the Subtraction Formula and use the Even and Odd Trigonometric Properties (Definition 1.5.22) where

\sin (- β) = - \sin β

and

\cos (- β) = \cos β

to get

\begin{aligned} \cos α \cos (- β) + \sin α \sin (- β) & = \cos (α - (- β)) \\ \cos α \cos (β) + \sin α (- \sin β) & = \cos (α + β) \\ \cos α \cos β - \sin α \sin β & = \cos (α + β) \end{aligned}

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Example 3.2.2.

Find the exact value of

\cos 105^{\circ} .

Solution.

First note that

105^{\circ} = 60^{\circ} + 45^{\circ} .

Now, using the Addition Formula for Cosine (Definition 3.2.1),

\begin{aligned} \cos 105^{\circ} & = \cos (60^{\circ} + 45^{\circ}) \\ = \cos 60^{\circ} \cos 45^{\circ} - \sin 60^{\circ} \sin 45^{\circ} \\ = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \frac{\sqrt{2}}{2} \\ = \frac{1}{4} (\sqrt{2} - \sqrt{6}) \end{aligned}

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Example 3.2.3.

Find the exact value of

\cos (\frac{π}{4} - \frac{π}{6}) .

Solution.

Using the Subtraction Formula for Cosine we get

\begin{aligned} \cos (\frac{π}{4} - \frac{π}{6}) & = \cos \frac{π}{4} \cos \frac{π}{6} + \sin \frac{π}{4} \sin \frac{π}{6} \\ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \\ = \frac{1}{4} (\sqrt{6} + \sqrt{2}) \end{aligned}

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Example 3.2.4.

Find the exact value of the expression

\cos 25^{\circ} \cos 35^{\circ} - \sin 25^{\circ} \sin 35^{\circ} .

Solution.

Notice this expression is the Addition Formula for Cosine with

α = 25^{\circ}

and

β = 35^{\circ} .

\cos 25^{\circ} \cos 35^{\circ} - \sin 25^{\circ} \sin 35^{\circ} = \cos (25^{\circ} + 35^{\circ}) = \cos 60^{\circ} = \frac{1}{2}

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Subsection 3.2.2 Addition and Subtraction Formulas for Sine

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Next we will learn about the addition and subtraction formulas for the sine function.

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Definition 3.2.5. Addition and Subtraction Formulas for Sine.

\begin{aligned} \sin (α + β) & = \sin α \cos β + \cos α \sin β \\ \sin (α - β) & = \sin α \cos β - \cos α \sin β \end{aligned}

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We will prove the Addition Formula for Sine in Example 3.2.10 and the Subtraction Formula can be established using the Even and Odd Properties (Definition 1.5.22).

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Example 3.2.6.

Given

\sin α = - \frac{12}{13},

with

\frac{3 π}{2} < α < 2 π

and

\cos β = - \frac{3}{5},

with

\frac{π}{2} < β < π,

find the exact value of

\sin (α + β) .

Solution.

The Addition Formula for Sine gives us

\sin (α + β) = \sin α \cos β + \cos α \sin β

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At this moment, we do not know the exact values of

\cos α

and

\sin β

but we can compute them.

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Given

\sin α = - \frac{12}{13}

with

\frac{3 π}{2} < α < 2 π

and

\cos β = - \frac{3}{5}

with

\frac{π}{2} < β < π,

we can draw the following triangles associated with

α

and

β,

respectively:

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Next, using the Pythagorean Theorem, we solve for the missing sides on the triangles

\begin{aligned} x^{2} + (- 12)^{2} & = 13^{2} \\ x + 144 & = 169 \\ x^{2} & = 25 \\ x & = 5 \end{aligned}

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Thus we get

\cos α = \frac{5}{13}

\begin{aligned} (- 3)^{2} + y^{2} & = 5^{2} \\ 9 + y^{2} & = 25 \\ y^{2} & = 16 \\ y & = 4 \end{aligned}

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Thus we get

\sin β = - \frac{4}{5}

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We now have all the information needed to proceed.

\begin{aligned} \sin (α + β) & = \sin α \cos β + \cos α \sin β \\ = (- \frac{12}{13}) (- \frac{3}{5}) + (\frac{5}{13}) (\frac{4}{5}) \\ = \frac{36}{65} + \frac{20}{65} \\ = \frac{56}{65} \end{aligned}

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Notice we did not have to know the values of

α

β

to do this example.

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Subsection 3.2.3 Addition and Subtraction Formulas for Tangent

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Now we will learn about the addition and subtraction formulas for the tangent function.

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Definition 3.2.7. Addition and Subtraction Formulas for Tangent.

\begin{aligned} \tan (α + β) & = \frac{\tan α + \tan β}{1 - \tan α \tan β} \\ \tan (α - β) & = \frac{\tan α - \tan β}{1 + \tan α \tan β} \end{aligned}

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Proof.

Recall that

\tan θ = \frac{\sin θ}{\cos θ}

as long as

\cos θ \neq 0 .

Using this fact, and our new formulas for the sum of sine and cosine, we get

\begin{aligned} \tan (α + β) & = \frac{\sin (α + β)}{\cos (α + β)} \\ = \frac{\sin α \cos β + \cos α \sin β}{\cos α \cos β - \sin α \sin β} \\ = \frac{\frac{\sin α \cos β + \cos α \sin β}{\cos α \cos β}}{\frac{\cos α \cos β - \sin α \sin β}{\cos α \cos β}} \\ = \frac{\frac{\sin α \cos β}{\cos α \cos β} + \frac{\cos α \sin β}{\cos α \cos β}}{\frac{\cos α \cos β}{\cos α \cos β} - \frac{\sin α \sin β}{\cos α \cos β}} \\ = \frac{\frac{\sin α}{\cos α} + \frac{\sin β}{\cos β}}{1 - \frac{\sin α}{\cos α} \frac{\sin β}{\cos β}} \\ = \frac{\tan α + \tan β}{1 - \tan α \tan β} \end{aligned}

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The subtraction formula can be established using the Even and Odd Properties (Definition 1.5.22).

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Example 3.2.8.

Find the exact value of

\tan (\frac{3 π}{4} + \frac{π}{6})

Solution.

Using the Addition Formula for Tangent (Definition 3.2.7), we get

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\begin{aligned} \tan (\frac{3 π}{4} + \frac{π}{6}) & = \frac{\tan \frac{3 π}{4} + \tan \frac{π}{6}}{1 - \tan \frac{3 π}{4} \tan \frac{π}{6}} \\ = \frac{(- 1) + \frac{\sqrt{3}}{3}}{1 - (- 1) \cdot \frac{\sqrt{3}}{3}} \\ = \frac{- 1 + \frac{\sqrt{3}}{3}}{1 + \frac{\sqrt{3}}{3}} \\ = \frac{\frac{- 3 + \sqrt{3}}{3}}{\frac{3 + \sqrt{3}}{3}} \\ = \frac{- 3 + \sqrt{3}}{3 + \sqrt{3}} \end{aligned}

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Subsection 3.2.4 Cofunction Identities

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Recall the Cofunction Identities (Definition 1.4.7):

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\begin{aligned} \sin θ & = \cos (\frac{π}{2} - θ), & \cos θ & = \sin (\frac{π}{2} - θ) \\ \tan θ & = \cot (\frac{π}{2} - θ), & \cot θ & = \tan (\frac{π}{2} - θ) \\ \sec θ & = \csc (\frac{π}{2} - θ), & \csc θ & = \sec (\frac{π}{2} - θ) \end{aligned}

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Armed with the knowledge of the subtraction formulas, we can prove the Cofunction Identities.

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Proof.

We will prove the Cofunction Identity for

\sin θ

in Example 3.2.9. The proof for

\cos θ

is given as Exercise 3.2.7.97. The Cofunction Identities for

\tan θ

and

\cot θ

can be found using the Quotient Identities (Definition 1.4.3) for

\csc θ

and

\sec θ

can be found using the Reciprocal Identities (Definition 1.4.2).

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Example 3.2.9.

Use the Subtraction Formula for Sine to establish the identity

\cos θ = \sin (\frac{π}{2} - θ) .

Solution.

To establish an identity, we will start from one side of the equality and use properties to end up with the expression on the other side of the equality. So,

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\begin{aligned} \sin (\frac{π}{2} - θ) & = \sin (\frac{π}{2}) \cos (θ) - \cos (\frac{π}{2}) \sin (θ) \\ = 1 \cdot \cos (θ) - 0 \cdot \sin (θ) \\ = \cos (θ) \end{aligned}

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Visually, we have

\cos θ = \frac{x}{r} = \sin (\frac{π}{2} - θ) :

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Example 3.2.10.

Prove the Addition Formula for Sine

\sin (α + β) = \sin α \cos β + \cos α \sin β

Solution.

We begin by using the cofunction identity

\begin{aligned} \sin (α + β) & = \cos (\frac{π}{2} - (α + β)) \\ = \cos ((\frac{π}{2} - α) - β) \end{aligned}

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By the Subtraction Formula for Cosine:

\begin{aligned} = \cos (\frac{π}{2} - α) \cos β + \sin (\frac{π}{2} - α) \sin β \\ = \sin α \cos β + \cos α \sin β \end{aligned}

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where the last step we use the Cofunction Identity.

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Subsection 3.2.5 Sums of Sines and Cosines

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Sometimes we may come across functions of the form

a \sin x + b \cos x

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It can often be useful to rewrite this expression as a single trigonomteric function.

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Definition 3.2.11.

For any real numbers

a

and

b,

let

θ

be an angle in standard position where

P (a, b)

is a point on the terminal side of

θ .

Then

a \sin x + b \cos x = \sqrt{a^{2} + b^{2}} \sin (x + θ) .

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Proof.

We begin by considering the triangle formed by the angle

θ

and point

P (a, b),

shown in Figure 3.2.12. By the Pythagorean Theorem, the hypotenuse of this triangle, with base

a

and height

b,

\sqrt{a^{2} + b^{2}} .

According to Definition 1.4.1, we have:

\cos θ = \frac{a}{\sqrt{a^{2} + b^{2}}}, \sin θ = \frac{b}{\sqrt{a^{2} + b^{2}}}

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or equivalently:

a = \sqrt{a^{2} + b^{2}} \cos θ, b = \sqrt{a^{2} + b^{2}} \sin θ

Figure 3.2.12. A triangle is formed by angle $θ$ and point $P (a, b) .$

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Therefore, using the addition formula for sine, we get

\begin{aligned} a \sin x + b \cos x & = \sqrt{a^{2} + b^{2}} \cos θ \sin x + \sqrt{a^{2} + b^{2}} \sin θ \cos x \\ = \sqrt{a^{2} + b^{2}} (\cos θ \sin x + \sin θ \cos x) \\ = \sqrt{a^{2} + b^{2}} \sin (x + θ) \end{aligned}

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Example 3.2.13.

Express

- \frac{\sqrt{3}}{2} \sin x + \frac{1}{2} \cos x

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in terms of sine only.

Solution.

To express the given expression in terms of sine only, we will use Definition 3.2.11. Considering the point

P (a, b) = (- \frac{\sqrt{3}}{2}, \frac{1}{2}),

which lies in Quadrant II, we determine the angle

θ .

Using either Table 1.5.18 or inverse trigonometric methods where

\tan θ = \frac{b}{a},

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we find

θ = 150^{\circ} .

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Therefore, by Definition 3.2.11, we have:

\begin{aligned} - \frac{\sqrt{3}}{2} \sin x + \frac{1}{2} \cos x & = \sqrt{{(- \frac{\sqrt{3}}{2})}^{2} + {(\frac{1}{2})}^{2}} \sin (x + 150^{\circ}) \\ = \sqrt{\frac{3}{4} + \frac{1}{4}} \sin (x + 150^{\circ}) \\ = \sin (x + 150^{\circ}) \end{aligned}

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Subsection 3.2.6 Summary

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To review, the addition and subtraction formulas are

\begin{aligned} \sin (α + β) & = \sin α \cos β + \cos α \sin β \\ \sin (α - β) & = \sin α \cos β - \cos α \sin β \\ \cos (α + β) & = \cos α \cos β - \sin α \sin β \\ \cos (α - β) & = \cos α \cos β + \sin α \sin β \\ \tan (α + β) & = \frac{\tan α + \tan β}{1 - \tan α \tan β} \\ \tan (α - β) & = \frac{\tan α - \tan β}{1 + \tan α \tan β} \end{aligned}

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Exercises 3.2.7 Exercises

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Exercise Group.

Use the Addition or Subtraction Formula to find the exact value of each expression.

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1.

\cos (15^{\circ})

Section 3.2 Addition and Subtraction Formulas

Subsection 3.2.1 Addition and Subtraction Formulas for Cosine

Definition 3.2.1. Addition and Subtraction Formulas for Cosine.

Proof.

Example 3.2.2.

Solution.

Example 3.2.3.

Solution.

Example 3.2.4.

Solution.

Subsection 3.2.2 Addition and Subtraction Formulas for Sine

Definition 3.2.5. Addition and Subtraction Formulas for Sine.

Example 3.2.6.

Solution.

Subsection 3.2.3 Addition and Subtraction Formulas for Tangent

Definition 3.2.7. Addition and Subtraction Formulas for Tangent.

Proof.

Example 3.2.8.

Solution.

Subsection 3.2.4 Cofunction Identities

Proof.

Example 3.2.9.

Solution.

Example 3.2.10.

Solution.

Subsection 3.2.5 Sums of Sines and Cosines

Definition 3.2.11.

Proof.

Example 3.2.13.

Solution.

Subsection 3.2.6 Summary

Exercises 3.2.7 Exercises

Exercise Group.

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Exercise Group.

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