If sin theta = -4/5 and theta is in quadrant 4, the value of sec theta

Mathematics

2 answers:

katovenus [111]3 years ago

4 0

Sec theta would be 5/3

Citrus2011 [14]3 years ago

4 0

Answer:

$\dfrac{5}{3}$

Step-by-step explanation:

We know that $\sec{\theta}=\dfrac{1}{\cos{\theta}}$ . On the other hand,

we can use the equality

$\sin^2 \theta + \cos^2 \theta=1$

to find the absolute value of $\cos \theta$ . It holds that

$\left[\dfrac{4}{5}\right]^2+\cos^2 \theta = 1 \Rightarrow \\\\\\\quad \cos^2 \theta = 1-\dfrac{16}{25}=\dfrac{9}{25} \Rightarrow\\ \\\cos \theta = \pm \dfrac{3}{5}$

Now, in the quadrant 4 the values of cosine is positive for all theta, hence

$\cos{\theta}=\dfrac{3}{5} \quad \Rightarrow \quad \sec \theta=\dfrac{1}{\cos\theta}=\dfrac{5}{3}$

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Que is on pic.i can't able to type in text.

ad-work [718]

It's not difficult to compute the values of $A$ and $B$ directly:

$A=\displaystyle\int_1^{\sin\theta}\frac{\mathrm dt}{1+t^2}=\tan^{-1}t\bigg|_{t=1}^{t=\sin\theta}$
$A=\tan^{-1}(\sin\theta)-\dfrac\pi4$

$B=\displaystyle\int_1^{\csc\theta}\frac{\mathrm dt}{t(1+t^2)}=\int_1^{\csc\theta}\left(\frac1t-\frac t{1+t^2}\right)\,\mathrm dt$
$B=\left(\ln|t|-\dfrac12\ln|1+t^2|\right)\bigg|_{t=1}^{t=\csc\theta}$
$B=\ln\left|\dfrac{\csc\theta}{\sqrt{1+\csc^2\theta}}\right|+\dfrac12\ln2$

Let's assume $0$ , so that $|\csc\theta|=\csc\theta$ .

Now,

$\Delta=\begin{vmatrix}A&A^2&B\\e^{A+B}&B^2&-1\\1&A^2+B^2&-1\end{vmatrix}$
$\Delta=A\begin{vmatrix}B^2&-1\\A^2+B^2&-1\end{vmatrix}-e^{A+B}\begin{vmatrix}A^2&B\\A^2+B^2&-1\end{vmatrix}+\begin{vmatrix}A^2&B\\B^2&-1\end{vmatrix}$
$\Delta=A(-B^2+A^2+B^2)-e^{A+B}(-A^2-A^2B-B^3)+(-A^2-B^3)$
$\Delta=A^3-A^2-B^3+e^{A+B}(A^2+A^2B+B^3)$

There doesn't seem to be anything interesting about this result... But all that's left to do is plug in $A$ and $B$ .

3 0

3 years ago

What is mathematics?

RUDIKE [14]

Answer:

Mathematics is a branch of science which involves numbers,space and quantity.

Step-by-step explanation:

I hope this helps you in any way.Have a great day :)

8 0

3 years ago

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Atalien is 5 feet tall. His shadow is 8 feet long. At the same time of day, a tree's shadow casts a 32 feet long shadow. What is