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vivado [14]
2 years ago
15

Rewrite \sqrt((1+cos45)/(2)) using a half-angle identity

Mathematics
1 answer:
aleksklad [387]2 years ago
7 0

\stackrel{\textit{Half-Angle Identities}}{cos\left(\cfrac{\theta}{2}\right)=\pm \sqrt{\cfrac{1+cos(\theta)}{2}}} \\\\[-0.35em] ~\dotfill\\\\ \sqrt{\cfrac{1+cos(45^o)}{2}}~~ = ~~cos\left( \cfrac{45^o}{2} \right)\implies \sqrt{\cfrac{1+cos(45^o)}{2}}~~ = ~~cos(22.5^o)

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Which is equivalent to tan θ?
Aloiza [94]

That which is equivalent to tanθ is A sinθ/cosθ

To answer the question, we need to know what tanθ is

<h3>What is tanθ?</h3>

Tanθ is a trigonometric identity which is the tangent of the angle .

Now, from the unit circle, with radius, r = 1, we have that tanθ = y/x.

Also, the sine of the angle θ is sinθ = x/1 = x

And also, the cosine of the angle θ is cosθ = y/1 = y

Since we have that

  • tanθ = y/x,
  • sinθ = x and
  • cosθ = y

Substituting the values of the variables y and x into tanθ, we have

tanθ = y/x

tanθ = sinθ/cosθ

So, that which is equivalent to tanθ is A sinθ/cosθ

Learn more about tanθ here:

brainly.com/question/26609988

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5 0
2 years ago
Emma is training for a 10-kilometer race. She wants to beat her last 10-kilometer time, which was 1 hour 10 minutes. Emma has al
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let x be how much longer she can run the race and still beat her previous time

55(min) + x(min) < 1(hr) + 10(min)

55(min) + x(min) < 60(min) + 10(min)

55 + x < 60 + 10

x < 60 + 10 - 55

x < 15(min)

7 0
3 years ago
The Smith family is driving to a vacation spot. The number of miles they have left is 540-60x, where x represents the number of
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Answer:

C

Step-by-step explanation:

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Does the line y = 3x + 1 pass through the point (4,15)
Firdavs [7]
No. when the line reaches 4 on the x axis it is at 13 on the y axis.
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2 years ago
Find all possible values of α+
const2013 [10]

Answer:

\rm\displaystyle  0,\pm\pi

Step-by-step explanation:

please note that to find but α+β+γ in other words the sum of α,β and γ not α,β and γ individually so it's not an equation

===========================

we want to find all possible values of α+β+γ when <u>tanα+tanβ+tanγ = tanαtanβtanγ</u><u> </u>to do so we can use algebra and trigonometric skills first

cancel tanγ from both sides which yields:

\rm\displaystyle  \tan( \alpha )  +  \tan( \beta ) =  \tan( \alpha )  \tan( \beta )  \tan( \gamma )  -  \tan( \gamma )

factor out tanγ:

\rm\displaystyle  \tan( \alpha )  +  \tan( \beta ) =   \tan( \gamma ) (\tan( \alpha )  \tan( \beta ) -  1)

divide both sides by tanαtanβ-1 and that yields:

\rm\displaystyle   \tan( \gamma ) =  \frac{ \tan( \alpha )  +  \tan( \beta ) }{ \tan( \alpha )  \tan( \beta )    - 1}

multiply both numerator and denominator by-1 which yields:

\rm\displaystyle   \tan( \gamma ) =   -  \bigg(\frac{ \tan( \alpha )  +  \tan( \beta ) }{ 1 - \tan( \alpha )  \tan( \beta )   } \bigg)

recall angle sum indentity of tan:

\rm\displaystyle   \tan( \gamma ) =   -  \tan( \alpha  +  \beta )

let α+β be t and transform:

\rm\displaystyle   \tan( \gamma ) =   -  \tan( t)

remember that tan(t)=tan(t±kπ) so

\rm\displaystyle   \tan( \gamma ) =    -\tan(   \alpha   +\beta\pm k\pi )

therefore <u>when</u><u> </u><u>k </u><u>is </u><u>1</u> we obtain:

\rm\displaystyle   \tan( \gamma ) =    -\tan(   \alpha   +\beta\pm \pi )

remember Opposite Angle identity of tan function i.e -tan(x)=tan(-x) thus

\rm\displaystyle   \tan( \gamma ) =    \tan(   -\alpha  -\beta\pm \pi )

recall that if we have common trigonometric function in both sides then the angle must equal which yields:

\rm\displaystyle  \gamma  =      -   \alpha   -  \beta \pm \pi

isolate -α-β to left hand side and change its sign:

\rm\displaystyle \alpha  +  \beta  +   \gamma  =  \boxed{ \pm \pi  }

<u>when</u><u> </u><u>i</u><u>s</u><u> </u><u>0</u>:

\rm\displaystyle   \tan( \gamma ) =    -\tan(   \alpha   +\beta \pm 0 )

likewise by Opposite Angle Identity we obtain:

\rm\displaystyle   \tan( \gamma ) =    \tan(   -\alpha   -\beta\pm 0 )

recall that if we have common trigonometric function in both sides then the angle must equal therefore:

\rm\displaystyle  \gamma  =      -   \alpha   -  \beta \pm 0

isolate -α-β to left hand side and change its sign:

\rm\displaystyle \alpha  +  \beta  +   \gamma  =  \boxed{ 0  }

and we're done!

8 0
3 years ago
Read 2 more answers
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