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Precalculus : Sum and Difference Identities For Tangent

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Example Question #1 : Sum And Difference Identities

In the problem below, $\sin \alpha = \frac {1}{5}, 0 < \alpha < \frac {\pi}{2}$ and $\cos \beta = \frac {1}{4}, 0 < \beta< \frac {\pi}{2}$ .

Find

$\tan (\alpha + \beta)$ .

Possible Answers:

$\frac {4}{5}$

$\frac {25\sqrt{6}-32\sqrt{15}}{9}$

$\frac {32\sqrt{6}-25\sqrt{15}}{9}$

$\frac {25\sqrt{6}+32\sqrt{15}}{9}$

$\frac {32\sqrt{6}+25\sqrt{15}}{9}$

Correct answer:

$\frac {32\sqrt{6}+25\sqrt{15}}{9}$

Explanation:

Since $\sin \alpha = \frac {1}{5}$ and $\alpha$ is in quadrant I, we can say that y = 1 and r = 5 and therefore:

$x = \sqrt {r^2 - y^2} = \sqrt {25-1} = \sqrt {24} = 2 \sqrt {6}$ .

So
$\tan \alpha = \frac {1}{2\sqrt{6}} = \frac {\sqrt{6}}{12}$ .

Since $\cos \beta = \frac {1}{4}$ and $\beta$ is in quadrant I, we can say that x = 1 and r = 4 and therefore:

$y = \sqrt {r^2 - x^2} = \sqrt {16-1} = \sqrt {15}$ .

So $\tan \beta = \sqrt{15}$ .

Using the tangent sum formula, we see:

$\\ \tan(\alpha+\beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta} \\ \\ \\* = \frac{\frac{\sqrt{6}}{12}+\sqrt{15}}{1-\frac{\sqrt{6}}{12} \cdot \frac{\sqrt{15}}{1}}}} = \frac{\frac{\sqrt{6}+12\sqrt{15}}{12}}{\frac{12-3\sqrt{10}}{12}} \\ \\ \\* = \frac{\sqrt{6}+12\sqrt{15}}{12-3\sqrt{10}} \cdot \frac{12+3\sqrt{10}}{12+3\sqrt{10}} \\ \\ \\* = \frac{12\sqrt{6}+6\sqrt{15}+144\sqrt{15}+180\sqrt{6}}{144-90} \\ \\* = \frac{192\sqrt{6}+150\sqrt{15}}{54} = \frac{32\sqrt{6}+25\sqrt{15}}{9}$

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Example Question #2 : Sum And Difference Identities

In the problem below, $\sin \alpha = \frac {1}{5}, 0 < \alpha < \frac {\pi}{2}$ and $\cos \beta = \frac {1}{4}, 0 < \beta< \frac {\pi}{2}$ .

Find

$\tan(\alpha - \beta)$ .

Possible Answers:

$\frac {25\sqrt{6}-32\sqrt{15}}{9}$

$\frac{1}{9}$

$\frac {25\sqrt{6}+32\sqrt{15}}{9}$

$\frac {32\sqrt{6}+25\sqrt{15}}{9}$

$\frac {32\sqrt{6}-25\sqrt{15}}{9}$

Correct answer:

$\frac {32\sqrt{6}-25\sqrt{15}}{9}$

Explanation:

Since $\sin \alpha = \frac {1}{5}$ and $\alpha$ is in quadrant I, we can say that y = 1 and r = 5 and therefore:

$x = \sqrt {r^2 - y^2} = \sqrt {25-1} = \sqrt {24} = 2 \sqrt {6}$ .

So $\tan \alpha = \frac {1}{2\sqrt{6}} = \frac {\sqrt{6}}{12}$ .

Since $\cos \beta = \frac {1}{4}$ and $\beta$ is in quadrant I, we can say that x = 1 and r = 4 and therefore:

$y = \sqrt {r^2 - x^2} = \sqrt {16-1} = \sqrt {15}$ .

So $\tan \beta = \sqrt{15}$ .

Using the tangent sum formula, we see:

$\\ \tan(\alpha-\beta) = \frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta} \\ \\ \\* = \frac{\frac{\sqrt{6}}{12}-\sqrt{15}}{1+\frac{\sqrt{6}}{12} \cdot \frac{\sqrt{15}}{1}}}} = \frac{\frac{\sqrt{6}-12\sqrt{15}}{12}}{\frac{12+3\sqrt{10}}{12}} \\ \\ \\* = \frac{\sqrt{6}-12\sqrt{15}}{12+3\sqrt{10}} \cdot \frac{12-3\sqrt{10}}{12-3\sqrt{10}} \\ \\ \\* = \frac{12\sqrt{6}-6\sqrt{15}-144\sqrt{15}+180\sqrt{6}}{144-90} \\ \\ \\* = \frac{192\sqrt{6}-150\sqrt{15}}{54} = \frac{32\sqrt{6}-25\sqrt{15}}{9}$

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Example Question #1 : Sum And Difference Identities

Given that $\tan x = 2$ and $\tan y = 3$ , find $\tan (x+y)$ .

Possible Answers:

Correct answer:

Explanation:

Jump straight to the tangent sum formula:

$\\ \tan (x+y) = \frac {\tan x + \tan y}{1-\tan x \tan y}$

From here plug in the given values and simplify.

$\\ \\ = \frac {2+3}{1-2\cdot3} \\ \\= \frac {5}{-5} \\ \\= -1$

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Example Question #1 : Sum And Difference Identities

Which of the following expressions best represents $\tan\left ( 60^{\circ}\right )$ ?

Possible Answers:

$\frac{2\tan(15^\circ)}{1-\tan^2(15^\circ)}$

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

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

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

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

Correct answer:

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

Explanation:

Write the identity for $tan(2\theta)$ .

$tan(2x)=\frac{2tan(x)}{1-tan^2(x)}$

Set the value of the angle equal to .

2x=60

x=30

Substitute the value of into the identity.

$\frac{2tan(30)}{1-tan^2(30)}$

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Precalculus : Sum and Difference Identities For Tangent

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Example Question #1 : Sum And Difference Identities

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Example Question #1 : Sum And Difference Identities

Example Question #1 : Sum And Difference Identities

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