Trigonometric Ratios

In the given figure, $AD=DB$, so find $\sin \theta , \cos \theta , \tan \theta$ and $sec \theta$

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

If $c=8$, then the area of the triangle is?

If $\sin \theta =\frac{12}{13}$, then evaluate $\left(\frac{2\sin \theta -3\cos \theta }{4\sin \theta -9\cos \theta }\right)$.

In a right angle triangle $ABC$, right-angled at $B$, if $\cos A= \frac{4}{5}$, then what is $\sin C$ equal to?

If $\tan \theta =\frac{1}{\sqrt{7}}$ then prove that $\left(\frac{\cos e{c}^2\theta +se{c}^2\theta }{\cos e{c}^2\theta -se{c}^2\theta }\right)=\frac{4}{3}$

If one side of a rhombus has end point $(4, 5)$ and $(1, 1)$, then maximum area of the rhombus is

$\frac{\tan A}{1-cotA}+\frac{cotA}{1-\tan A}=secA \cos ecA+1$

The height of a telephone pole from the ground is $17 \mathrm{m}$. The wire tied to its top and makes an angle of $45°$ with the ground. Fond the length of the wire.

In an equilateral $∆\mathrm{ABC}$, $\mathrm{AD}\perp \mathrm{BC}$ and ${\mathrm{AD}}^2:{\mathrm{BC}}^2=\mathrm{x}:1$, then $\mathrm{x}$ is equal to:

In the figure shown below, lines $AB$ and $\mathrm{PQ}$ are parallel to each other. All measurements are in centimetres.

Which of the following gives the value of $\cos \theta$?

The diagonal of a rectangle makes an angle $60°$ with each of the smaller sides. If the breadth of the rectangle is $2 \mathrm{cm}$, find the length of the diagonal (without units).

A $90 \mathrm{cm}$ wide door opens on one side of the room at a maximum angle of $90°$. Due to shortage of space, a $40 \mathrm{cm}$ by $80 \mathrm{cm}$ table is kept behind the door along the wall such that it obstructs its path.

(Note: The figure is not to scale)

At what distance from the hinge should the table be kept such that the door opens for a maximum angle of $60°$. Show your work. (Use $\sqrt{2}=1.41, \sqrt{3}=1.73$)

The teacher asked the students to correctly complete the following sentence about the rhombus.

"A rhombus has a side length of $l$ units and one of its angles is equal to $\mathrm{\theta }$. The ratio of the lengths of the two diagonals is dependent on _____."

Ashima: only $l$.

Bilal: only $\mathrm{\theta }$.

Chris: both $l$ and $\mathrm{\theta }$.

Duleep: neither $l$ or $\mathrm{\theta }$.

Who answered the question correctly? Show your work and give valid reasons.

Find $\mathrm{PA}$ (without units).

The sine of an angle in a right triangle is $\frac{4}{5}$.

Which of these could be the measures of the sides of the triangle? 

In the figure shown below, lines $AB\text{ and} PQ$ are parallel to each other. All measurements are in centimetres.

Which of the following gives the value of $\cos \theta$?

$\sin \left(90°\right)$ is

In $△ABC, \angle A=\theta , \angle B=90°$ and $\angle C=\varphi$.

If $BC=x$ and $AC=\left(x+2\right)$, find the value of $\cos \theta$.

In $△ABC, \angle A=\theta , \angle B=90°$ and $\angle C=\varphi$.

If $BC=x$ and $AC=\left(x+2\right)$, find the value of $\left(\sqrt{{\mathrm{x}}^3+x^2}\right) \mathrm{tan\theta }$.

In $△ABC, \angle A=\theta , \angle B=90°$ and $\angle C=\varphi$.

If $BC=x$ and $AC=\left(x+2\right)$, find the value of $\left(\sqrt{\mathrm{x}+1}\right)\cot \mathrm{\varphi }$.