Basic Trigonometric Identities

Q.

If $\frac{sinx}{siny}=\frac{1}{2},\frac{cosx}{cosy}=\frac{3}{2},x,yϵ(0,\frac{\pi }{2})$, then tan (x+y)=___

1. $\sqrt{17}$

2. $\sqrt{15}$

3. $\sqrt{14}$

4. $\sqrt{13}$

Q.

The value of x for the maximum value of $\sqrt{3}cosx+sinx$ is

1. 30⁰

2. 60⁰

3. 90⁰

4. 45⁰

Q.

Differentiate $sin2xcos3x.$

Q.

If sinx + cosx = $\frac{1}{5}$ then the value of cos2x

1. 

2. 

3. 

4. 

Q. Prove that:
$\cos 4x=1-8{\cos }^{2}x+8{\cos }^{4}x$

Q.

Find the general value of x for which $\sqrt{3}$ cosec x =2.

Q.

If sinx + cosx = $\frac{1}{5}$ then find the value of sinx.cosx.

1. 

2. 

3. 

4. 

Q.

S = ${\sum }_{r=0}^{10}co{s}^3\frac{r\pi }{3}$ find the value of 16S

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Q.

Given sinx + cosx = $\frac{5}{4}$. If 1 + 2sinxcosx = a, find the value of 32 a.

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Q. 1. cosx--lies in third quadrant.

Q.

$\text{In a triangle ABC,}\frac{cosA}{a}=\frac{cosB}{b}=\frac{cosC}{c}\text{If a=}\frac{1}{\sqrt{6}}\text{then the area of the triangle (in square unit) is}$

1. √3/24

2. 1/√3

3. 1/8

4. 1/24

Q.

If 3 cosx+ 2cos3x = cosy, 3sinx+2sin3x=siny, then the value of $4(cos2x{)}^2$ is

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Q.

$If\frac{2sin\alpha }{1+cos\alpha +sin\alpha }=y,then\frac{1-cos\alpha +sin\alpha }{1+sin\alpha }isequalto$

1. 

2. 

3. 

4. 

Q. If $\sin \theta +{\sin }^2\theta =1,$ then ${\cos }^2\theta +{\cos }^4\theta =$

1. \N
2. $\sqrt{2}$
3. 1
4. 2

Q.

If arg(z) = ${\log }_e{i}^i$, then the value of complex number z is

1. purely imaginary

2. Purely real

3. Has both real & imaginary parts

4. Can't say

Q. Prove that }\operatorname{tan}(θ+135^°)\operatorname{tan}(θ-315^°)=-1

Q. ${\tan }^4\theta +{\tan }^2\theta$ is equal to

1. ${\sec }^2-{\sec }^4\theta$
2. ${\sec }^4\theta -{\sec }^2\theta$
3. ${\sin }^4\theta -{\sec }^2\theta$
4. ${\sin }^4\theta -{\cos }^2\theta$

Q. A sector $OABO$ of central angle $\theta$ is constructed in a circle with centre $O$ and radius $6.$ The radius of the circle that is circumscribed about the triangle $OAB,$ is

1. $6\cos \frac{\theta }{2}$
2. $6\sec \frac{\theta }{2}$
3. $6(\cos \frac{\theta }{2}+2)$
4. $3\sec \frac{\theta }{2}$

Q.

If sin2A = sin(90-3A). Find the value of sinA

1. 

2. 

3. 1

4. 

Q.

Prove that:

$2{\text{sin}}^2\frac{pi}{6}+{\text{cosec}}^2\frac{7\pi }{6}{\text{cos}}^2\frac{\pi }{3}=\frac{3}{2}$

Q. sin A + sin B = $\sqrt{3}$ (cos B – cos A) $\Rightarrow$ sin 3A + sin 3B =

1. 0
2. 2
3. 1
4. -1

Q. 135. If cos(a+2b)=m cosa then How To prove that (m-1)cotb=(m+1)tanb

Q. If $\sin \theta ,\cos \theta$ are the roots of the equation $a{x}^2-bx+c=0,$ then the relation among a, b, c is

1. $a^2-b^2+2ac=0$
2. $a^2+b^2-2ac=0$
3. $a^2-b^2-2ac=0$
4. $a^2+b^2+2ac=0$

Q. Given $\sin x+\cos x=\frac{5}{4}$. If $1+2\sin x\cos x=a,$ then $32a=$

1. 10
2. 32
3. 50
4. 1

Q.

The minimum value of 3cosx+4sinx+5 is

[MNR 1991]

1. 5

2. 9

3. 0

4. 7

Q.

Differentiate x cos x by first principle.

Or

Evaluate $li{m}_{y\to 0}\frac{(x+y)sec(x+y)-xsecx}{y}$

Q.

10 $co{s}^{2}x$ - 6 sinxcosx + 2$si{n}^{2}x$ is equals to

1. (A) 6 + 4 cos2x + 3sin2x

2. (A) 6 - 4 cos2x + 3sin2x

3. 6 + 4 cos2x - 3 sin2x

4. 6 - 4 cos2x - 3sin2x

Q. , x in quadrant II

Q.

$Ifaco{s}^3\alpha +3acos\alpha si{n}^2\alpha =mandasi{n}^3\alpha +3aco{s}^2\alpha sin\alpha =n,then(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}isequalto$

1. 

2. 

3. 

4. 

Q.

y = $\frac{si{n}^2{1500}^{\circ }+4si{n}^4{750}^{\circ }-4si{n}^2{750}^{\circ }co{s}^2{750}^{\circ }}{4-si{n}^2{1500}^{\circ }+4si{n}^2{750}^{\circ }}$ Find the value of 9y.

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