Trigonometric Ratios Using Right Angled Triangle

Q.

Determine the smallest positive value of x (in degrees) for which tan $(x+{100}^{\circ })=tan(x+{50}^{\circ })tanxtan(x-{50}^{\circ })$.

Prove that $\frac{cos7x-cos8x}{1+2cos5x}=cos2x-cos3x$

Q.

The least positive integer n for which ${\left(\frac{1+i}{1-i}\right)}^n=\frac{2}{\pi }(se{c}^{-1}\frac{1}{x}+si{n}^{-1}x),x\ne 0,-1\le x\le 1$ is:

1. 2

2. 6

3. 8

4. 4

Q. An aeroplane flying at a height of $9000\text{m}$ vertically above from the ground, passes another aeroplane when the angles of elevation of the two aeroplanes from a point on the ground are ${60}^{\circ }$ and ${30}^{\circ }$ respectively. Then the vertical distance between the aeroplanes at that instant is

1. $1000\text{m}$
2. $3000\text{m}$
3. $6000\text{m}$
4. $8000\text{m}$

Q. 30. A)SinA+sinB=a, cosA+cosB=b, find , 1)Sin (A+B)

Q.

Match the following:

Given sin x = $\frac{2}{5}$ and x $ϵ$ $(0,\frac{\Pi }{2})$

$\begin{array}{cc}\text{(p) cos x}& \text{(1)}\frac{5}{2}\\ \text{(q) tan x}& \text{(2)}\frac{\sqrt{21}}{2}\\ \text{(r) cosec x}& \text{(3)}\frac{\sqrt{21}}{5}\\ & \text{(4)}\frac{2}{\sqrt{21}}\end{array}$

1. P-3, Q-2, R-4

2. P-2, Q-3, R-1

3. None of these

4. P-2, Q-3, R-4

Q. In triangle ABC, $AB=5,BC=12,AC=13,\angle ACB=x$ and $\angle ABC={90}^{\circ }.$ If the value of $\sin x+\sec x+\tan x$ is $a,$ then $156a=$

1. 394
2. 384
3. 294
4. 400

Q. Prove that:
${\sin }^{2}24°-{\sin }^{2}6°=\frac{\sqrt{5}-1}{8}$

Q.

If the value of sinx + secx + tanx is a, find 156a.

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Q. In any $\Delta ABC$, the least value of $\left(\frac{si{n}^{2}A+sinA+1}{sinA}\right)$ is

1. $\sqrt{3}$
2. 3
3. None of these
4. 9

Q.

Let $A_0{A}_1{A}_2{A}_3{A}_4{A}_5$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $A_0{A}_1,A_0{A}_{2}and{A}_0{A}_4$ is

1. 

2. 

3. 

4. 

Q. If 3 sin θ + 4 cos θ = 5, then 4 sin θ − 3 cos θ =

(a) 0
(b) 5
(c) 1
(d) None of these

Q.

Match the following:

Given sin x = $\frac{2}{5}$ and x $ϵ$ $(0,\frac{\Pi }{2})$

(p) cosx (1) $\frac{5}{2}$

(q) tanx (2) $\frac{\sqrt{21}}{2}$

(r) cosecx (3) $\frac{\sqrt{21}}{5}$

(4) $\frac{2}{\sqrt{21}}$

1. None of these

2. P-2, Q-3, R-4

3. P-2, Q-3, R-1

4. P-3, Q-2, R-4

Q. If $3\tan \left(x-15°\right)=\tan \left(x+15°\right)$, $0<x<90°$, find $\theta$.

Q.

In $\Delta ABC,$ prove that

$\frac{(b-c)}{b+c}=\frac{tan\frac{1}{2}(B-C)}{tan\frac{1}{2}(B+C)}$

Q. Given, $\sin x=\frac{2}{5}$ and $x\in (0,\frac{\pi }{2})$

Match the following:

1. $\frac{2}{\sqrt{21}}$
2. $\frac{\sqrt{21}}{5}$
3. $\frac{\sqrt{21}}{2}$
4. $\frac{5}{\sqrt{21}}$

Q. 2) If the roots of the quadratic equation :
$$\begin{gathered} x^2+Ax+B=0are\tan 30°and\tan 15°thenthevalueofA-B=— \\ \left(a\right)1 \\ \left(b\right)-1 \\ \left(c\right)2 \\ \left(d\right)3 \end{gathered}$$

Q.

Let $A_0{A}_1{A}_2{A}_3{A}_4{A}_5$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $A_0{A}_1,A_0{A}_{2}and{A}_0{A}_4$ is

1. $\frac{3}{4}$

2. $3\sqrt{3}$

3. $3$

4. $\frac{3\sqrt{3}}{2}$

Q. Prove that cos π / 15 . cos 2 π / 15 . cos 3 π / 15 . cos 4 π / 15 . cos 5 π / 15 . cos 6 π / 15 . cos 7 π / 15 = 1/ 2^7

Q. $\frac{{\cos }^{3}A+{\sin }^{3}A}{\cos A+\sin A}+\frac{{\cos }^{3}A-{\sin }^{3}A}{\cos A-\sin A}$

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

Q. A tower $T_1$ of height $60$ m is located exactly opposite to a tower $T_2$ of height $80m$ on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_2$, then the width (in $m$) of the road between the feet of the towers $T_1$ and $T_2$ is:

1. $10\sqrt{3}$
2. $10\sqrt{2}$
   
3. $20\sqrt{2}$
4. $20\sqrt{3}$

Q. Prove that:
$\cos \frac{\pi }{15}\cos \frac{2\pi }{15}\cos \frac{3\pi }{15}\cos \frac{4\pi }{15}\cos \frac{5\pi }{15}\cos \frac{6\pi }{15}\cos \frac{7\pi }{15}=\frac{1}{128}$

Q. Reduce each of the following expressions to the sine and cosine of a single expression:

(i) $\sqrt{3}\sin \mathrm{\theta }-\cos \mathrm{\theta }$
(ii) cos θ − sin θ
(iii) 24 cos θ + 7 sin θ

Q. If $\sin \alpha =\frac{4}{5}\mathrm{and}\cos \beta =\frac{5}{13}$, prove that $\cos \frac{\alpha -\beta }{2}=\frac{8}{\sqrt{65}}$

Q.

find the value of cospi/8 , sinpi/8 , tanpi/8 , sinpi/24 , cospi/24 , tan(142x1/2)degree

Q.

cos 47 cos13 - sin 47 sin13