7.2 Addition and Subtraction Formulas

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7.2.1 Learning Objectives

By the end of this section, you should be able to:

• apply sine, cosine, and tangent addition formulas;
• apply subtraction formulas;
• calculate exact values for nonstandard angles;
• simplify expressions using cofunction relationships; and
• verify identities involving sums and differences.

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7.2.2 Sine and Cosine Formulas

\[ \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta, \]

\[ \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta. \]

The sign in the cosine formula changes.

Example 7.4 Finding an Exact Sine Value

Find \(\sin75^\circ\).

Solution

\[ \begin{aligned} \sin75^\circ &=\sin(45^\circ+30^\circ)\\ &=\frac{\sqrt2}{2}\frac{\sqrt3}{2} +\frac{\sqrt2}{2}\frac12\\ &=\frac{\sqrt6+\sqrt2}{4}. \end{aligned} \]

\[ \boxed{\sin75^\circ=\frac{\sqrt6+\sqrt2}{4}} \]

Exercise 7.4 Find \(\cos15^\circ\).

Show answer

Answer: \(\boxed{(\sqrt6+\sqrt2)/4}\)

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7.2.3 Tangent Formulas

\[ \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}, \]

\[ \tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}. \]

Example 7.5 Finding an Exact Tangent Value

Find \(\tan75^\circ\).

Solution

\[ \tan75^\circ =\frac{1+1/\sqrt3}{1-1/\sqrt3} =2+\sqrt3. \]

\[ \boxed{2+\sqrt3} \]

Exercise 7.5 Find \(\tan15^\circ\).

Show answer

Answer: \(\boxed{2-\sqrt3}\)

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7.2.4 Applying Known Values

When \(\alpha\) and \(\beta\) are not standard angles, determine missing signs or values from their quadrants and identities before applying a formula.

Example 7.6 Using Two Known Ratios

If \(\sin\alpha=3/5\) in Quadrant I and \(\cos\beta=-12/13\) in Quadrant II, find \(\cos(\alpha+\beta)\).

Solution

\[ \cos\alpha=4/5,\qquad \sin\beta=5/13. \]

\[ \cos(\alpha+\beta)=\frac45\left(-\frac{12}{13}\right)-\frac35\left(\frac5{13}\right) =-\frac{63}{65}. \]

\[ \boxed{-63/65} \]

Exercise 7.6 If \(\cos\alpha=4/5\) and \(\sin\beta=12/13\), with both angles acute, find \(\sin(\alpha+\beta)\).

Show answer

Answer: \(\sin\alpha=3/5\), \(\cos\beta=5/13\), so \(\boxed{63/65}\).

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7.2.5 Conceptual Takeaways

• A trigonometric function of a sum is not the sum of the function values.
• Exact values can be built from familiar angles.
• Quadrants determine missing signs.
• The cosine formula reverses the sign between terms.
• Addition and subtraction formulas support later double-angle identities.

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7.2.6 Skills You Should Be Able to Do

• Apply all addition and subtraction formulas.
• Decompose angles into standard angles.
• Calculate exact values.
• Determine missing ratios from quadrant information.
• Verify sum-and-difference identities.

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7.2.7 Practice Problems with Solutions

1. Find \(\sin105^\circ\).

   Show Solution

   \[ \sin(60^\circ+45^\circ)=\boxed{\frac{\sqrt6+\sqrt2}{4}}. \]

2. Find \(\cos75^\circ\).

   Show Solution

   \[ \cos(45^\circ+30^\circ)=\boxed{\frac{\sqrt6-\sqrt2}{4}}. \]

3. Find \(\sin15^\circ\).

   Show Solution

   \[ \sin(45^\circ-30^\circ)=\boxed{\frac{\sqrt6-\sqrt2}{4}}. \]

4. Find \(\tan105^\circ\).

   Show Solution

   \[ \tan(60^\circ+45^\circ)=\frac{\sqrt3+1}{1-\sqrt3}=\boxed{-(2+\sqrt3)}. \]

5. Verify \(\cos(x+\pi/2)=-\sin x\).

   Show Solution

   \[ \cos x\cos(\pi/2)-\sin x\sin(\pi/2)=0-\sin x. \]

   Thus, \(\boxed{\cos(x+\pi/2)=-\sin x}\).

6. If \(\sin\alpha=5/13\) and \(\cos\beta=3/5\), with both angles acute, find \(\sin(\alpha-\beta)\).

   Show Solution

   \(\cos\alpha=12/13\) and \(\sin\beta=4/5\). Therefore,

   \[ \sin(\alpha-\beta)=\frac5{13}\frac35-\frac{12}{13}\frac45 =\boxed{-33/65}. \]

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