Angles

In the triangle $ABC$, $a^2+b^2+c^2=ac+\sqrt{3}bc$

What is the nature of the triangle?

Evaluate: $\cot \left({\sin }^{-1}\frac{1}{\sqrt{5}}+{\sin }^{-1}\frac{2}{\sqrt{5}}\right)$

Let $\overrightarrow{a}=3\hat{i}+3\hat{j}+3\hat{k}$ and $\overrightarrow{c}=\hat{j}-\hat{k}$. Let $\overrightarrow{b}$ be such that $\overrightarrow{a}·\overrightarrow{b}=27$ and $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{9c}$

What is the angle between $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and $\overrightarrow{c}$?

Consider two lines whose direction ratios are $\left(2, -1, 2\right)$ and $\left(k, 3, 5\right)$. They are inclined at an angle $\frac{\pi }{4}$.

What is the value of $k$?

If $z=\frac{1+i\sqrt{3}}{1-i\sqrt{3}}$ where $i=\sqrt{-1}$, then what is the argument of $z$?

In the $△ABC$, if $a=7, b=6 \text{and} A={120}^{\circ },$ then the approximate value of $B$ is

Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be three vectors such that

$\left|\overrightarrow{a}\right|=\sqrt{31}, 4\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=2 \text{and} 2\left(\overrightarrow{a}\times \overrightarrow{b}\right)=3\left(\overrightarrow{c}\times \overrightarrow{a}\right).$

If the angle between $\overrightarrow{b} \text{and} \overrightarrow{c} \text{is} \frac{2\mathrm{\pi }}{3}, \text{then} {\left(\frac{\overrightarrow{a}\times \overrightarrow{c}}{\overrightarrow{a}.\overrightarrow{b}}\right)}^{2 } \text{is equal to} ?$

Consider ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1.$ Let $C$ be centre of ellipse and $P$ is a variable point lying on the ellipse. If the angle between $CP$ and tangent at $P$ is minimum, then $P$ may be

Find the value of $\theta$ if $2\sin \left(2\theta \right)=\sqrt{3}$

Find the value of $\sin \frac{41\mathrm{\pi }}{4}$.

Find ${\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)+{\cos }^{-1}\frac{1}{2}=?$

$\sin ²140⁰+\cos ²40°$

If $\overrightarrow{a} \text{and} \overrightarrow{b}$ are unit vectors and $\overrightarrow{a}-\overrightarrow{b}$ is also a unit vector, then write the measure of the angle between $\overrightarrow{a} \text{and }\overrightarrow{b}.$

If ${\mathrm{m}}^2\cos \left(\frac{2\mathrm{\pi }}{15}\right)\cos \left(\frac{4\mathrm{\pi }}{15}\right)\cos \left(\frac{8\mathrm{\pi }}{15}\right)\cos \left(\frac{14\mathrm{\pi }}{15}\right)={\mathrm{n}}^2$ then find the value of $\frac{{\mathrm{m}}^2-{\mathrm{n}}^2}{{\mathrm{n}}^2}$

In $△ABC$, the value of $\cos \theta =$

Reduce $63°{14}^{\text{'}}{51}^{\text{'}\text{'}}$ to centesimal measure

Calculate the angle of $1°\left(\text{degree}\right), 1\text{'}$(minute of arc or arcmin) and $1"$(second of arc or arc second) in radians. Use $360°= 2\pi \text{ rad}, 1°= 60\text{'} \text{and} 1\text{'}= 60"$.

If $\overrightarrow{a}, \overrightarrow{b}$ be two vectors such that $\left|\overrightarrow{a}\right|=\frac{1}{2}$ and $\left|\overrightarrow{b}\right|=1.$ If $2\overrightarrow{a}-\overrightarrow{b}$ is a unit vector, then the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is$?$

Calculate the angle of  $1\text{'}\text{'} (\mathrm{Second} \mathrm{of} \mathrm{arc} \mathrm{or} \mathrm{arc} \mathrm{second})$ in radians
Use $360°=2\mathrm{\pi } \mathrm{rad}, 1°=60\text{'}, \mathrm{and} 1\text{'}=60\text{'}\text{'}.$$$

Calculate the angle of $1\text{'} (\mathrm{minute} \mathrm{of} \mathrm{arc} \mathrm{or} \mathrm{arcmin})$
Use $360°=2\mathrm{\pi } \mathrm{rad}, 1°=60\text{'}, \mathrm{and} 1\text{'}=60\text{'}\text{'}.$$$