Question #5a090

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Question #5a090

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Answer 1 · 2017-02-03T20:54:08

Probably not.

Explanation:

$a r c sec \sqrt{2} = \frac{π}{4} \approx 0.7854$ $arc sec sqrt2 = pi/4 ~~ 0.7854$ $" "$ (since $cos (\frac{π}{4}) = \frac{1}{\sqrt{2}}$ $cos (pi/4) = 1/sqrt2$ )

$a r c csc \sqrt{2} = \frac{π}{4} \approx 0.7854$ $arc csc sqrt2 = pi/4 ~~ 0.7854$ $" "$ (since $sin (\frac{π}{4}) = \frac{1}{\sqrt{2}}$ $sin (pi/4) = 1/sqrt2$ )

But $a r c cot 2$ $arc cot 2$ is not a rational multiple of $π$ $pi$ . You'll need a decimal approximation method to get

$a r c cot 2 \approx 0.4636$ $arc cot 2 ~~ 0.4636$

$3 (0.7854) - 4 (0.7854) + 5 (0.4636) = 1.5326 \approx 1.533$ $3(0.7854) -4(0.7854) +5(0.4636) = 1.5326 ~~ 1.533$

Answer 2 · 2017-02-03T21:39:51

$y = 5 {cot}^{- 1} (2) - \frac{π}{4}$ $y=5cot^-1(2)-pi/4$

Explanation:

Although this was in the "Differentiating Trigonometric Functions" section, it looks like you just want to evaluate or simplify what you provided.

$y = 3 {sec}^{- 1} (\sqrt{2}) - 4 {csc}^{- 1} (\sqrt{2}) + 5 {cot}^{- 1} (2)$ $y=3sec^-1(sqrt(2))-4csc^-1(sqrt(2))+5cot^-1(2)$

FIRST TERM: ${sec}^{- 1} (\sqrt{2})$ $sec^-1(sqrt(2))$
If $x = {sec}^{- 1} (\sqrt{2})$ $x=sec^-1(sqrt(2))$ , then $sec (x) = \sqrt{2}$ $sec(x)=sqrt(2)$ or $\frac{1}{cos (x)} = \sqrt{2}$ $1/cos(x)=sqrt(2)$ .
This is the same as saying

$cos (x) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ $cos(x)=1/sqrt(2)=sqrt(2)/2$ , which happens when $x = \frac{π}{4}$ $x=pi/4$

SECOND TERM: ${csc}^{- 1} (\sqrt{2})$ $csc^-1(sqrt(2))$
If $x = {csc}^{- 1} (\sqrt{2})$ $x=csc^-1(sqrt(2))$ , then $csc (x) = \sqrt{2}$ $csc(x)=sqrt(2)$
Identically, $\frac{1}{sin (x)} = \sqrt{2}$ $1/sin(x)=sqrt(2)$ or $sin (x) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ $sin(x)=1/sqrt(2)=sqrt(2)/2$ , which also happens at $x = \frac{π}{4}$ $x=pi/4$

THIRD TERM: ${cot}^{- 1} (2)$ $cot^-1(2)$
If $x = {cot}^{- 1} (2)$ $x=cot^-1(2)$ , then $cot (x) = 2$ $cot(x)=2$ or $\frac{1}{tan (x)} = 2$ $1/tan(x)=2$ or $tan (x) = \frac{1}{2}$ $tan(x)=1/2$ . Because the function $y = tan (x)$ $y=tan(x)$ repeats periodically (see graph), you actually get an infinite number of possible $x$ $x$ values. So you can't really say what ${cot}^{- 1} (2)$ $cot^-1(2)$ is.

graph{y=tan(x) [-20,20,-0.5,1]}

Most calculators assume you meant the middle one though and will give you an answer ${cot}^{- 1} (2) \approx 0.4636$ $cot^-1(2)~~0.4636$

ANSWER
Plugging those values in, you get
$y = 3 (\frac{π}{4}) - 4 (\frac{π}{4}) + 5 {cot}^{- 1} (2)$ $y=3(pi/4)-4(pi/4)+5cot^-1(2)$
$y = 5 {cot}^{- 1} (2) - \frac{π}{4}$ $y=5cot^-1(2)-pi/4$

A calculator will deceptively say $y \approx 1.5326$ $y~~1.5326$

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Question #5a090

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