8.1 Sum and Difference Formulas

Show that $\sin ({60}^{\circ }+{30}^{\circ })$ is not equal to $\sin ({60}^{\circ })+\sin ({30}^{\circ })$.

Find an exact value for $\sin ({75}^{\circ })$.

Suppose that $\sin (\theta )={\displaystyle \frac{-5}{13}}$ and $\cos (\theta )={\displaystyle \frac{12}{13}}$. Find an exact value for $\cos ({\displaystyle \frac{\pi }{4}}+\theta )$.

Evaluate $\sin \left({\displaystyle \frac{\pi }{12}}\right)$ exactly.

Evaluate $\tan \left({\displaystyle \frac{\pi }{12}}\right)$ exactly.

Find $\cos (2\theta )$ and $\tan (2\theta )$ for the angle $\theta$ shown in the previous Example.

For the triangle in the previous example, find expressions for $\sin (2\varphi )$ and $\tan (2\varphi )$.

Find an expression for $\cos (2\alpha )$ if you know that $\sin (\alpha )={\displaystyle \frac{6}{w}}$.

Solve $\cos (2t)=\cos t$ for $0\le x\le 2\pi$.

1. Sketch an angle $\alpha$ in standard position, with ${\displaystyle \frac{\pi }{2}}<\alpha <\pi$. Also sketch the angle $-\alpha$.
2. Choose a point on the terminal side of $\alpha$, and show that the negative angle identities hold for $\alpha$.

1. Sketch an angle $\beta$ in standard position, with $\pi <\beta <{\displaystyle \frac{3\pi }{2}}$. Also sketch the angle $-\beta$.
2. Choose a point on the terminal side of $\beta$, and show that the negative angle identities hold for $\beta$.

Given that $\sin \left({\displaystyle \frac{7\pi }{12}}\right)={\displaystyle \frac{\sqrt{2}+\sqrt{6}}{4}}$, find $\sin \left({\displaystyle \frac{-7\pi }{12}}\right)$. Sketch both angles.

Given that $\cos \left({\displaystyle \frac{7\pi }{12}}\right)={\displaystyle \frac{\sqrt{2}-\sqrt{6}}{4}}$, find $\cos \left({\displaystyle \frac{-7\pi }{12}}\right)$. Sketch both angles.

If $\cos (2x-0.3)=0.24$ and $\sin (2x-0.3)<0$, find $\cos (0.3-2x)$ and $\sin (0.3-2x)$.

If $\sin (1.5-\varphi )=-0.28$ and $\cos (1.5-\varphi )>0$, find $\sin (\varphi -1.5)$ and $\cos (\varphi -1.5)$.

Show that $\cos ({45}^{\circ }+{45}^{\circ })$ is not equal to $\cos ({45}^{\circ })+\cos ({45}^{\circ })$.

Show that $\tan ({60}^{\circ }-{30}^{\circ })$ is not equal to $\tan ({60}^{\circ })-\tan ({30}^{\circ })$.

Use your calculator to verify that $\tan ({87}^{\circ }-{29}^{\circ })$ is not equal to $\tan ({87}^{\circ })-\tan ({29}^{\circ })$.

Use your calculator to verify that $\cos ({52}^{\circ }+{64}^{\circ })$ is not equal to $\cos ({52}^{\circ })+\cos ({64}^{\circ })$.

Use graphs to show that $\sin (x-{\displaystyle \frac{\pi }{6}})$ is not equivalent to $\sin (x)-\sin \left({\displaystyle \frac{\pi }{6}}\right)$.

Use graphs to show that $\tan (x+{\displaystyle \frac{\pi }{4}})$ is not equivalent to $\tan x+\tan {\displaystyle \frac{\pi }{4}}$.

Suppose $$\cos (\alpha )={\displaystyle \frac{3}{5}},\sin (\alpha )={\displaystyle \frac{4}{5}},\cos (\beta )={\displaystyle \frac{5}{13}}$$, and $\sin (\beta )={\displaystyle \frac{-12}{13}}$. Evaluate the following.

1. $\cos (\alpha +\beta )$
2. $\sin (\alpha +\beta )$
3. $\tan (\alpha +\beta )$

Suppose $$\cos (\alpha )={\displaystyle \frac{-2}{3}},\sin (\alpha )={\displaystyle \frac{\sqrt{5}}{3}},\cos (\beta )={\displaystyle \frac{\sqrt{3}}{2}}$$, and $\sin (\beta )={\displaystyle \frac{-1}{2}}$. Evaluate the following.

1. $\cos (\alpha -\beta )$
2. $\sin (\alpha -\beta )$
3. $\tan (\alpha -\beta )$

If $\tan (t)={\displaystyle \frac{3}{4}}$ and $\tan (s)={\displaystyle \frac{-7}{24}}$, find exact values for:

1. $\tan (s+t)$
2. $\tan (s-t)$

If $\tan (x)=-3$ and $\tan (y)=-5$, find exact values for:

1. $\tan (x+y)$
2. $\tan (x-y)$

Suppose $\cos (\theta )={\displaystyle \frac{15}{17}}$ and $\sin (\varphi )={\displaystyle \frac{3}{5}}$, where $\theta$ and $\varphi$ are in quadrant I. Evaluate the following.

1. $\cos (\theta +\varphi )$
2. $\tan (\theta -\varphi )$

Suppose $\cos (\theta )={\displaystyle \frac{15}{17}}$, where $\theta$ is in quadrant IV, and $\sin (\varphi )={\displaystyle \frac{3}{5}}$, where $\varphi$ is in quadrant II. Evaluate the following.

1. $\sin (\theta -\varphi )$
2. $\tan (\theta +\varphi )$

If $\sin (\alpha )={\displaystyle \frac{12}{13}},{\displaystyle \frac{\pi }{2}}<\alpha <\pi$, and $\cos (\beta )={\displaystyle \frac{-3}{5}},\pi <\beta <{\displaystyle \frac{3\pi }{2}}$, find exact values for:

1. $\sin (\alpha +\beta )$
2. $\cos (\alpha +\beta )$
3. $\tan (\alpha +\beta )$
4. Sketch the angles $\alpha ,\beta$ and $\alpha +\beta$.

If $\cos (\alpha )={\displaystyle \frac{3}{8}},{\displaystyle \frac{3\pi }{2}}<\alpha <2\pi$, and $\sin (\beta )={\displaystyle \frac{-1}{4}},\pi <\beta <{\displaystyle \frac{3\pi }{2}}$, find exact values for:

1. $\sin (\alpha -\beta )$
2. $\cos (\alpha -\beta )$
3. $\tan (\alpha -\beta )$
4. Sketch the angles $\alpha ,\beta$ and $\alpha -\beta$.

Find the exact values of $\cos ({15}^{\circ })$ and $\tan ({15}^{\circ })$.

Find the exact values of $\sin ({165}^{\circ })$ and $\tan ({165}^{\circ })$.

If $\sin (\theta )=0.2$ and $\cos (\theta )>0$, find $\sin (\theta +{\displaystyle \frac{\pi }{3}})$.

If $\cos (\theta )=0.6$ and $\sin (\theta )<0$, find $\cos (\theta +{\displaystyle \frac{3\pi }{4}})$.

$$\sin (\theta -{270}^{\circ })$$

$$\cos ({270}^{\circ }+\theta )$$

$$\cos (t+{\displaystyle \frac{\pi }{6}})$$

$$\sin (t-{\displaystyle \frac{2\pi }{3}})$$

$$\tan (\beta -{\displaystyle \frac{\pi }{6}})$$

$$\tan (\varphi +{\displaystyle \frac{\pi }{4}})$$

Does $\sin (2\cdot {80}^{\circ })=2\sin ({80}^{\circ })$?

Does $\cos (2\cdot {25}^{\circ })=2\cos ({25}^{\circ })$?

Does $\tan (2\cdot {70}^{\circ })=2\tan ({70}^{\circ })$?

Does $\tan (2\cdot {100}^{\circ })=2\tan ({100}^{\circ })$?

$$\sin ({90}^{\circ })=2\sin ({45}^{\circ })\cos ({45}^{\circ })$$

$$\sin ({60}^{\circ })=2\sin ({30}^{\circ })\cos ({30}^{\circ })$$

$$\cos ({60}^{\circ })={\cos }^2({30}^{\circ })-{\sin }^2({30}^{\circ })$$

$$\tan ({60}^{\circ })={\displaystyle \frac{2\tan ({30}^{\circ })}{1-{\tan }^2({30}^{\circ })}}$$

If $\cos (\alpha )=0.32$, then $\cos (2\alpha )=2(0.32)=0.64$.

If $\cos (2\beta )=0.86$, then $\cos (\beta )=0.43$, so $\beta ={\cos }^{-1}(0.43)$.

If $\sin (2\theta )=h$, then $\sin (\theta )={\displaystyle \frac{h}{2}}$, so $\theta ={\sin }^{-1}\left({\displaystyle \frac{h}{2}}\right)$.

If $\cos (\varphi )=r$, then $\cos (2\varphi )=2r$.

$$2\sin ({34}^{\circ })\cos ({34}^{\circ })$$

$${\cos }^2\left({\displaystyle \frac{\pi }{10}}\right)-{\sin }^2\left({\displaystyle \frac{\pi }{10}}\right)$$

$$1-2{\sin }^2\left({\displaystyle \frac{\pi }{16}}\right)$$

$$2{\cos }^2({18}^{\circ })-1$$

$${\cos }^2(3\theta )-{\sin }^2(3\theta )$$

$$2\sin (2\alpha )\cos (2\alpha )$$

$$2\sin (5t)\cos (5t)$$

$${\cos }^2(4w)-{\sin }^2(4w)$$

$${\displaystyle \frac{2\tan ({64}^{\circ })}{1-{\tan }^2({64}^{\circ })}}$$

$${\displaystyle \frac{2\tan \left({\displaystyle \frac{\pi }{3}}\right)}{1-{\tan }^2\left({\displaystyle \frac{\pi }{3}}\right)}}$$

$$2{\cos }^2(2\beta )-1$$

$$1-2{\sin }^2(6s)$$

1. $\sin (\alpha )$
2. $\cos (\alpha )$
3. $\tan (\alpha )$
4. $\sin (2\alpha )$
5. $\cos (2\alpha )$
6. $\tan (2\alpha )$

1. $\sin (\beta )$
2. $\cos (\beta )$
3. $\tan (\beta )$
4. $\sin (2\beta )$
5. $\cos (2\beta )$
6. $\tan (2\beta )$

1. $\sin (s)$
2. $\cos (s)$
3. $\tan (s)$
4. $\sin (2s)$
5. $\cos (2s)$
6. $\tan (2s)$

1. $\sin (t)$
2. $\cos (t)$
3. $\tan (t)$
4. $\sin (2t)$
5. $\cos (2t)$
6. $\tan (2t)$

Suppose $\cos (\theta )={\displaystyle \frac{12}{13}}$ and ${\displaystyle \frac{3\pi }{2}}<\theta <2\pi$. Compute exact values for:

1. $\sin (\theta )$
2. $\sin (2\theta )$
3. $\cos (2\theta )$
4. $\tan (2\theta )$
5. Sketch the angles $\theta$ and $2\theta$.

Suppose $\sin (\varphi )={\displaystyle \frac{5}{6}}$ and ${\displaystyle \frac{\pi }{2}}<\varphi <\pi$. Compute exact values for:

1. $\cos (\varphi )$
2. $\sin (2\varphi )$
3. $\cos (2\varphi )$
4. $\tan (2\varphi )$
5. Sketch the angles $\varphi$ and $2\varphi$.

If $\tan (u)=-4$ and ${270}^{\circ }<u<{360}^{\circ }$, find exact values for:

1. $\tan (2u)$
2. $\cos (2u)$
3. $\sin (2u)$

If $\tan (v)={\displaystyle \frac{2}{3}}$ and ${180}^{\circ }<v<{270}^{\circ }$, find exact values for:

1. $\tan (2v)$
2. $\cos (2v)$
3. $\sin (2v)$

$$\sin (2\theta )+\sqrt{2}\cos (\theta )=0$$

$$\sin (2\alpha )\sin (\alpha )=\cos (\alpha )$$

$$\cos (2t)-5\cos (t)+3=0$$

$$\cos (2x)+3\sin (x)=2$$

$$\tan (2\beta )+2\sin (\beta )=0$$

$$\tan (2z)-2\cos (z)=0$$

$$3\cos (\varphi )-\sin ({\displaystyle \frac{\pi }{2}}-\varphi )=\sqrt{3}$$

$$\sin (w)+\cos ({\displaystyle \frac{\pi }{2}}-w)=1$$

$$\sin (2\varphi )\cos (\varphi )+\cos (2\varphi )\sin (\varphi )=1$$

$$\cos (\theta )\cos (3\theta )+\sin (\theta )\sin (3\theta )={\displaystyle \frac{\sqrt{2}}{2}}$$

Use the sum of angles formulas for sine and cosine to derive a formula for each expression. Then use graphs to verify your formula.

1. $\cos (\theta +{90}^{\circ })$
2. $\sin (\theta +{90}^{\circ })$

Use the sum of angles formulas for sine and cosine to derive a formula for each expression. Then use graphs to verify your formula.

1. $\cos (\theta +\pi )$
2. $\sin (\theta +\pi )$

Use the difference of angles formulas for sine and cosine to prove that:

1. $\cos ({\displaystyle \frac{\pi }{2}}-\theta )=\sin (\theta )$
2. $\sin ({\displaystyle \frac{\pi }{2}}-\theta )=\cos (\theta )$

Use the difference of angles formulas for sine and cosine to derive formulas for:

1. $\cos (\theta -{\displaystyle \frac{\pi }{2}})$
2. $\sin (\theta -{\displaystyle \frac{\pi }{2}})$

Prove the double angle identity $\sin (2\theta )=2\sin (\theta )\cos (\theta )$. (Hint: Start with the sum of angles formula for sine and replace both $\alpha$ and $\beta$ by $\theta$.)

Prove the double angle identity $$\cos (2\theta )={\cos }^2(\theta )-{\sin }^2(\theta )$$. (Hint: Start with the sum of angles formula for sine and replace both $\alpha$ and $\beta$ by $\theta$.)

$$\sin ({\displaystyle \frac{\pi }{2}}+\beta )=1+\sin (\beta )$$

$$\cos ({\displaystyle \frac{\pi }{3}}-\beta )=\cos (\beta -{\displaystyle \frac{\pi }{3}})$$

$$\sin (A+{180}^{\circ })=-\sin (A)$$

$$\tan (\theta )+\tan (-\theta )=0$$

$$\cos (4\theta )=4\cos (\theta )$$

$$\cos (\varphi +{\displaystyle \frac{\pi }{3}})={\displaystyle \frac{1}{2}}+\cos (\varphi )$$

$$\sin (x+{\displaystyle \frac{\pi }{4}})={\displaystyle \frac{\sqrt{2}}{2}}(\sin (x)+\cos (x))$$

$$2\cos (x-{\displaystyle \frac{\pi }{6}})=\sin (x)+\sqrt{3}\cos (x)$$

$$\sin (x-{\displaystyle \frac{\pi }{3}})+\cos (x+{\displaystyle \frac{\pi }{6}})=0$$

$$\cos (2x)=(\cos (x)+\sin (x))(\cos (x)-\sin (x))$$

The figure below shows a right triangle inscribed in a rectangle.

1. Label the legs $l_1$ and $l_2$ of the right triangle with their lengths.
2. Explain why ${\theta }_1=\beta$ and ${\theta }_2=\alpha +\beta$. Label the diagram with these angles.
3. Label the legs $s_1$ and $s_2$ of the bottom triangle with their lengths.
4. Label the legs $s_3$ and $s_4$ of the top left triangle with their lengths.
5. Label the legs $s_5$ and $s_6$ of the top right triangle with their lengths.
6. Use the fact that the opposite sides of a rectangle are equal to state the addition formulas for sine and cosine.

The figure below shows a right triangle inscribed in a rectangle.

1. Label the legs $l_1$ and $l_2$ of the right triangle with their lengths.
2. Explain why ${\theta }_1=\beta$ and ${\theta }_2=\alpha -\beta$. Label the diagram with these angles.
3. Label the legs $s_1$ and $s_2$ of the bottom triangle with their lengths.
4. Label the legs $s_3$ and $s_4$ of the top left triangle with their lengths.
5. Label the legs $s_5$ and $s_6$ of the top right triangle with their lengths.
6. Use the fact that the opposite sides of a rectangle are equal to state the subtraction formulas for sine and cosine.

Follow the steps to prove the difference of angles formula for cosine,

$$\cos (\alpha -\beta )=\cos (\alpha )\cos (\beta )+\sin (\alpha )\sin (\beta )$$

1. Write an expression for $(AB{)}^2$, the square of the distance between the points $A$ and $B$, using the Law of Cosines for $\mathrm{△}AOB$.
2. Write another expression for $(AB{)}^2$ using the distance formula and the coordinates of $A$ and $B$.
3. Equate the two expressions for $(AB{)}^2$ you obtained in (a) and (b). Simplify the equation to obtain $$\cos (\alpha -\beta )=\cos (\alpha )\cos (\beta )+\sin (\alpha )\sin (\beta )$$.