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) =

Sarcasticcarma Oct 28, 2022

#3**+2 **

Thanks Asinus,

Also asked and answered here

https://web2.0calc.com/questions/please-help-me-i-m-having-a-hard-time-understanding

Melody Oct 29, 2022

#2**+2 **

What is the exact value of sin(θ+β)?

**Hello Sarcasticcarma!**

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

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

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

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

!

asinus Oct 28, 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

#7**0 **

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**+2 **

Best Answer

Thanks Asinus,

Also asked and answered here

https://web2.0calc.com/questions/please-help-me-i-m-having-a-hard-time-understanding

Melody Oct 29, 2022