Trigonometry Questions

Find the minimum value of $5\sin \theta +12\cos \theta$.

If the length of shadow on the ground of a post is $\frac{1}{\sqrt{3}}$ times of its height, the angle of elevation of the Sun is:

If $\sin \theta =\frac{5}{13}$, then what is the value of $\tan \theta +\sec \theta$?

The height of a tower is $4h\text{meter}$ and the height of the observer is $h\text{meter}$. He finds the top of the tower from a distance $\sqrt{3}h\text{meter}$ from the tower. Find the angle of elevation of the tower as seen by the observer.

In a right-angled triangle $XYZ$, $\angle Y=90^{\circ }$, if $XY=2\sqrt{6}$ and $XZ-YZ=2$, then $\sec X+\tan X$ is

If $\cot \theta =\frac{15}{8}$, then what is the value of $\frac{(2+2\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(2-2\cos \theta )}$?

If $\csc \theta +\cot \theta =\sqrt{3}$, find the value of $\csc \theta$.

If $\sec \theta =\csc \varphi$ and $0^{\circ }<(\theta ,\varphi )<90^{\circ }$, then the value of $\sin (\theta +\varphi )$ is:

In $△ABC$, $\angle B=90^{\circ }$, $BC=7\text{cm}$. If $AC-AB=1\text{cm}$, then the value of $\sin A$ (denoted here as $\sin \theta$ for the targeted angle) is:

An arc of length $121\text{cm}$ subtends an angle of $77^{\circ }$ at the centre of a circle. The radius of the circle is:

If $3\sin \alpha +5\cos \alpha =5$, then the value of $(3\cos \alpha -5\sin \alpha {)}^2$ is equal to

If cos(
2
A
​
)=x, then the value of x is:

If 2csc
2
A=x, then the value of x is:

If tan(−
4
7π
​
)=x, then the value of x is:

If
cotB−cotA
cotAcotB+1
​
=x, then the value of x is:

If 2sec
2
A=x, then the value of x is:

The greatest and the least value of $a+b\sin \theta$ ($0^{\circ }\le \theta \le 90^{\circ }$) are $7$ and $3$ respectively. Find the values of $a$ and $b$.

If $\tan 2\theta \tan \theta =1$, then the value of $\tan 2\theta$ is:

If $\tan \theta +\sin \theta =m$ and $\tan \theta -\sin \theta =n$, then which one of the following relations is correct?

If cot(−
4
5π
​
)=x, then the value of x is:

The height of a tower is $4h\text{meters}$ and the height of the observer is $h\text{meters}$. He views the top of the tower from a horizontal distance of $3\sqrt{3}h\text{meters}$ from the base. Find the angle of elevation of the tower as seen by the observer.

If $\cot \theta =\frac{x}{y}$, then what is the value of $\frac{x\cos \theta -y\sin \theta }{x\cos \theta +y\sin \theta }$?

If the length of the shadow on the ground of a post is $\frac{1}{\sqrt{3}}$ times its height, the angle of elevation of the Sun is:

If $\tan \theta +\sec \theta =4$, then what is the value of $\cos \theta$?

Find the minimum value of $5\sin \theta +12\cos \theta$.

In a right-angled triangle $XYZ$, $\angle Y=90^{\circ }$. If $XY=2\sqrt{6}$ and $XZ-YZ=2$, then the value of $\sec X+\tan X$ is:

Two angles of a triangle are $75^{\circ }$ and $45^{\circ }$. What is the value of the third angle in circular measure? (As per convention, do not use the symbol of radian.)

Find the value of (sin75
∘
+sin15
∘
).

If ${\sin }^{-1}a+{\sin }^{-1}b+{\sin }^{-1}c=\pi$, then the value of $a\sqrt{1-a^2}+b\sqrt{1-b^2}+c\sqrt{1-c^2}$ will be:

If Psin60
∘
=Qcsc45
∘
, then the value of
P
2
−Q
2

P
2
+Q
2

​
is equal to:

If 3sec
4
θ+8=10sec
2
θ, then the value of tanθ can be:

If $\cot \theta =\frac{28}{45}$, then find the value of $(\sin \theta -\cos \theta )$.

If $\sec 6A=\csc (A-29^{\circ })$, where $2A$ is an acute angle, then find the measure of $\angle A$.

Simplify the expression: $(1+{\tan }^{2}A)+\left(1+\frac{1}{{\tan }^{2}A}\right)$

If $\tan (t)+\cot (t)=1$, then one of the values of the expression $\frac{1}{\sin (t)+\cos (t)}$ is:

The expression sin
2
θ+cos
2
θ−1=0 is satisfied by how many values of θ?

If ${\cos }^{-1}\left(\frac{3}{5}\right)-{\sin }^{-1}\left(\frac{4}{5}\right)={\cos }^{-1}x$, then $x=$

If 8tanA=5, what is the value of
8sinA+11cosA
8sinA−7cosA
​
?

If $\sin \alpha =\frac{5}{13}$, then the value of $\cos \alpha \cdot \csc \alpha \cdot \cot \alpha$ is _________.

Find the value of $(1+\cot A-\csc A)(1+\tan A+\sec A)-3({\sin }^{2}A+{\cos }^{2}A)$.

If $\sin x+\sin y=a$ and $\cos x+\cos y=b$, then find the value of $(\sin x\sin y+\cos x\cos y)$.

Find the value of $\csc \theta (1-\cos \theta )(\csc \theta +\cot \theta )$.

In ΔABC, the measure of angle B is 90
∘
. If sinA=
25
24
​
and AB=1.4cm, then what is the length (in cm) of side BC?

If tanθ+cotθ=2, then find the value of tan
53
θ+cot
38
θ.

Which of the following is NOT a correct formula?

Find the value of $\sin \left(\frac{\pi }{3}-{\sin }^{-1}(-\frac{1}{2})\right)$

If $x,y,z$ are in A.P. and ${\tan }^{-1}x,{\tan }^{-1}y$ and ${\tan }^{-1}z$ are also in A.P., then:

$\text{ABCD}$ is a trapezium such that $\text{AB}$ and $\text{CD}$ are parallel and $\text{BC}\perp \text{CD}$. If $\angle \text{ADB}=\theta$, $\text{BC}=p$ and $\text{CD}=q$, then $\text{AB}$ is equal to:

The expression $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$ can be written as:

In a $\Delta PQR$, if $3\sin P+4\cos Q=6$ and $4\sin Q+3\cos P=1$, then the angle $R$ is equal to:

If $A={\sin }^{2}x+{\cos }^{4}x$, then for all real $x$:

Let $\cos (\alpha +\beta )=\frac{4}{5}$ and let $\sin (\alpha -\beta )=\frac{5}{13}$, where $0\le \alpha ,\beta \le \frac{\pi }{4}$, then $\tan 2\alpha =$

Let $A$ and $B$ denote the statements:$A:\cos \alpha +\cos \beta +\cos \gamma =0$$B:\sin \alpha +\sin \beta +\sin \gamma =0$ If $\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=-\frac{3}{2}$, then:

$AB$ is a vertical pole with $B$ at the ground level and $A$ at the top. A man finds that the angle of elevation of the point $A$ from a certain point $C$ on the ground is $60^{\circ }$. He moves away from the pole along the line $BC$ to a point $D$ such that $CD=7\text{m}$. From $D$, the angle of elevation of the point $A$ is $45^{\circ }$. Then the height of the pole is:

If in a triangle $ABC$, $a{\cos }^2\left(\frac{C}{2}\right)+c{\cos }^2\left(\frac{A}{2}\right)=\frac{3b}{2}$, then the sides $a,b$ and $c$:

The sum of the radii of inscribed and circumscribed circles for an $n$ -sided regular polygon of side $a$, is:

The upper $\frac{3}{4}$ th portion of a vertical pole subtends an angle ${\tan }^{-1}\frac{3}{5}$ at point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is:

If in a triangle $ABC$, the altitudes from the vertices $A,B,C$ on opposite sides are in H.P., then $\sin A,\sin B,\sin C$ are in:

In a triangle $ABC$, let $\angle C=\frac{\pi }{2}$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then $2(r+R)$ equals:

If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}=\alpha$, then $4x^2-4xy\cos \alpha +y^2$ is equal to:

△XYZ is right-angled at Y. If m∠Z=45
∘
, then find the value of (
3

​

cscX−2
​
).

If cos60
∘
−sec30
∘
=x, then the value of x is:

If
2
1−cosA
​
=x, then the value of x is:

If sinA+cosA=
3
4
​
, then find the value of tanA+cotA.

△XYZ is right-angled at Y. If m∠Z=45
∘
, then find the value of (cscX−
3

​

2
​
).

In △PQR, the measure of angle Q is 90
∘
. If tanP=
3
4
​
and 1.5cm, then what is the length (in cm) of side PR?

If $\tan 45^{\circ }+\text{cosec }30^{\circ }=x$, then find the value of $x$.

If $\sqrt{\frac{1+\cos A}{2}}=x$, then find the value of $x$.

If $\sec A+\tan A=x$, then find the value of $x$.

If cscA−cotA=x, then the value of x is:

In △PQR, the measure of angle Q is 90
∘
. If tanP=
3
4
​
and PQ=1.5cm, then what is the length (in cm) of side PR?

If cosec
−1
(
6
11π
​
)=x, then what is the value of x?

If [
1+sinA
cosA
​
]
2
=x, then the value of x is:

If sec(−
4
7π
​
)=x, then the value of x is:

If tan(
2
A
​
)=x, then the value of x is:

If (secA−tanA)
2
=x, then the value of x is:

What is the value of (
2
1
​
−cot45
∘
)?

If cot(
2
A
​
)=x, then the value of x is:

If cos
2
θ−sin
2
θ−3cosθ+2=0, 0
∘
<θ<90
∘
, then what is the value of
2
1+
5

​
cosθ+tanθ
​
?

If 4tanA=3, then cos
2
A−sin
2
A equals:

If $2{\cos }^2\theta -5\cos \theta +2=0$, $0^{\circ }<\theta <90^{\circ }$, then the value of $\frac{1}{\csc \theta +\cot \theta }$ is:

If sinθ+cosθ=
3
11

​

​
, then what is tanθ+cotθ equal to?

The value of
sec
2
θ
32
​
−
1+tan
2
θ
20
​
+12sin
2
θ is:

If tan3θ=cot(θ−22
∘
), where 3θ is an acute angle, find the value of θ.

If 4sin
2
θ−4
3

​
sinθ+3=0, then find the value of θ (0
∘
<θ<90
∘
).

In a triangle $PQR$, $\angle R=\frac{\pi }{2}$. If $\tan \frac{P}{2}$ and $\tan \frac{Q}{2}$ are the roots of $a{x}^2+bx+c=0,a\ne 0$, then:

What is the value of $\left(\csc 60^{\circ }+\frac{1}{2}\right)$?

In $△DEF$, the measure of angle $E$ is $90^{\circ }$. If $\cos D=\frac{5}{13}$ and $DE=1\text{cm}$, then what is the length (in $\text{cm}$) of side $EF$?

If sin(−
3
5π
​
)=x, then the value of x is:

If
1−tanAtanB
tanA+tanB
​
=x, then the value of x is:

If
cscA−cotA
1
​
=x, then the value of x is:

If csc(−
3
4π
​
)=x, then the value of x is:

If
1+cosA
1−cosA
​
=x, then the value of x is:

If $f:R\to S,$ defined by $f(x)=\sin x-\sqrt{3}\cos x+1,$ is onto, then the interval of $S$ is

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $60^{\circ }$ and when he retires $40$ meter away from the tree the angle of elevation becomes $30^{\circ }.$ The breadth of the river is

The sides of a triangle are $\sin \alpha ,\cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha }$ for some $0<\alpha <\frac{\pi }{2}.$ Then the greatest angle of the triangle is

If $u=\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }+\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta },$ then the difference between the maximum and minimum values of $u^2$ is given by

Let $\alpha ,\beta$ be such that $\pi <\alpha -\beta <3\pi .$ If $\sin \alpha +\sin \beta =-\frac{21}{65}$ and $\cos \alpha +\cos \beta =-\frac{27}{65},$ then the value of $\cos \frac{\alpha -\beta }{2}$ is