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# cosθ=−√2/6 , where π≤θ≤3π/2 .

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cosθ=−√2/6 , where π≤θ≤3π/2 .

tanβ=5/12 , where 0≤β≤π/2 .

What is the exact value of sin(θ+β)?

Enter your answer, as a single fraction in simplified form, in the box.

sin(θ+β)  =

Oct 28, 2022

#3
+118132
+2
Oct 29, 2022

#2
+14085
+2

What is the exact value of sin(θ+β)?

Hello Sarcasticcarma!

$$cos\ θ=−\frac{\sqrt{2}}{6}\ |\ π\le θ\le\frac{3\pi}{2}\\ tan\ β=\frac{5}{12}\ |\ 0\le\beta\le \frac{\pi}{2}\\$$

$$sin(\theta+\beta)=sin \theta\ cos\beta+cos\theta\ sin\beta\\ sin\theta=-\sqrt{1-cos^2\theta}\ \Leftarrow\ 3rd\ quadrant\\ sin\beta=\frac{tan\beta}{\sqrt{1+tan^2\beta}}\ \\ cos\beta=\frac{1}{\sqrt{1+tan^2\beta}}\\ sin(\theta+\beta)=-\sqrt{1-cos^2\theta}\cdot \frac{1}{\sqrt{1+tan^2\beta}}+cos\theta\cdot \frac{tan\beta}{\sqrt{1+tan^2\beta}}\\$$

$$sin(\theta+\beta)=-\sqrt{1-(-\frac{\sqrt{2}}{6})^2}\cdot \dfrac{1}{\sqrt{1+(\frac{5}{12})^2}}+(-\frac{\sqrt{2}}{6})\cdot \dfrac{(\frac{5}{12})}{\sqrt{1+(\frac{5}{12})^2}}\\$$

$$sin(\theta+\beta)=-\frac{\sqrt{34}}{6}\ \cdot \ \frac{12}{13}\ -\ \frac{\sqrt{2}}{6}\ \cdot \ \frac{5}{ 13}\\ \color{blue}sin(\theta+\beta)=-\dfrac{12\cdot \sqrt{34}+5\cdot \sqrt{2}}{78}=-0.9877$$

!

Oct 28, 2022
edited by asinus  Oct 28, 2022
edited by asinus  Oct 28, 2022
edited by asinus  Oct 28, 2022
edited by asinus  Oct 28, 2022
edited by asinus  Oct 29, 2022
edited by asinus  Oct 29, 2022
edited by asinus  Oct 29, 2022
#4
+2

Hi asinus!
I think we got different answers as:

$$sin(\beta) \neq \dfrac{tan(\beta)}{\sqrt{1-tan^2(\beta)}}$$

But I think you meant: $$sin(\beta)=\dfrac{tan(\beta)}{\sqrt{1+tan^2(\beta)}}$$ which is an identity.

Also, $$sin(\theta) \neq \sqrt{1-cos^2(\theta)}$$ but rather, $$sin(\theta)=\pm \sqrt{1-cos^2(\theta)}$$, and we choose +ve or -ve sign depending on the quadrant.

Since theta is in the third quadrant, then sin(theta) should be negative, so we must choose the -ve, and not the positive, as then theta will be in the first quadrant or second, but we are given it is in the third.

Guest Oct 29, 2022
#5
+14085
+2

Thanks very much! I will use your hints.

!

asinus  Oct 29, 2022
#6
+1

:)

Guest Oct 29, 2022
#7
+22
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Thank you so much for your answer!! I just couldn't work it out for some reason and you doing it helped me a lot!

Sarcasticcarma  Nov 7, 2022
#3
+118132
+2