Trigonometric Ratios

Q. The string of a kite is 180 meters long and it makes an angle of 30º with the horizontal. If there is no slack in the string, then the height of the kite is

1. 90 m
2. $30\sqrt{3}m$
3. $60\sqrt{3}m$
4. $90\sqrt{3}m$

Q. Question 3
If tan (A + B) = $\sqrt{3}$ and tan (A - B) = $\frac{1}{\sqrt{3}}$; $0^{\circ }<A+B\le {90}^{\circ };A>B$, find A and B.

Q.

If tan $\theta =\frac{20}{21}$, show that $\frac{(1-sin\theta +cos\theta )}{(1+sin\theta +cos\theta )}$ = $\frac{3}{7}$.

Q.

A ladder rests against a wall at an angles $\alpha$ to the horizontal. Its foot is pulled away from the wall through a distance ${}^{\prime }{p}^{\prime }$, so that it slides a distance ${}^{\prime }{q}^{\prime }$ down the wall making an angle $\beta$ with the horizontal. Show that $\frac{a}{b}=\frac{cos\alpha -cos\beta }{sin\beta -sin\alpha }$

Q. In a right $∆\mathrm{ABC}$, right-angled at $\mathrm{B}$, if $\mathrm{tanA}=1$, then verify that $2\mathrm{sinA}·\mathrm{cosA}=1$.

Q.

Prove that sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A

Q.

If cosec $\theta =\sqrt{10}$, find the values of all T - ratios of $\theta$

Q.

The greatest and the least values of ${\left[{\sin }^{-1}\left(x\right)\right]}^3+{\left[{\cos }^{-1}\left(x\right)\right]}^3$ are

1. $\left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$

2. $\left(-\frac{{\mathrm{\pi }}^3}{8}\right)$

3. $\left(\frac{7{\mathrm{\pi }}^3}{8},\frac{{\mathrm{\pi }}^3}{32}\right)$

4. None of these

Q.

Calculate the value $\tan \left(9^0\right)-\tan \left({27}^0\right)-\tan \left({63}^0\right)+\tan \left({81}^0\right)$

Q.

The value of $\underset{x\to \infty }{\lim }{\left[\frac{\mathrm{\pi }}{2}-{\tan }^{-1}\left(x\right)\right]}^{\frac{1}{x}}$ is

1. $0$

2. $1$

3. $-1$

4. $e$

Q.

If $\sin \left(\theta \right)=\frac{7}{25}$ then find the values of $\cos \left(\theta \right),\tan \left(\theta \right)$.

Q.

If $\cos A=\frac{3}{4}$, then $32\sin \left(\frac{A}{2}\right)\sin \left(\frac{5A}{2}\right)=$

1. $7$

2. $8$

3. $11$

4. None of these

Q. Sin 53

Q.

If $\theta ={30}^o$, verify that:

(i) $tan2\theta =\frac{2tan\theta }{1-ta{n}^2\theta }$

(ii) $sin2\theta =\frac{2tan\theta }{1+ta{n}^2\theta }$

(iii) $cos2\theta =\frac{1-ta{n}^2\theta }{1+ta{n}^2\theta }$

(iv) $cos3\theta =4co{s}^3\theta -3cos\theta$

Q.

How is $\cos (-x)=\cos x$?

Q.

What is the maximum value of $\frac{1}{sec\theta }$?

Q.

If $x=acosec\theta$ and $y=bcot\theta$, then which of the following equations is true ?

1. $x^2-y^2=a^2-b^2$

2. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

3. $x^2+y^2=a^2+b^2$

4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Q. Prove D/dx(sinx)=cosx

Q.

If $\sin \left(A\right)-\cos \left(A\right)=0$, then find the value of ${\sin }^4\left(A\right)+{\cos }^4\left(A\right)$

1. $0$

2. $1$

3. $\frac{1}{2}$

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

Q.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

Q. Question 10
$cos\theta =\frac{a^2+b^2}{2ab}$, where a and b are two distinct numbers such that ab>0.

Q.

If sin⁡(A + B) = 1 and sin ⁡(A -B) = $\frac{1}{2}$, such that 0 ≤ A + B ≤ 90° and A > B, then find A and B.

Q.

(Cot A) is the product of cot and A.

1. True

2. False

Q. The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ − b cos θ) is

(a) a² + b²

(b) a + b

(c) a² − b²

(d) $\sqrt{a2+b2}$

Q.

If $\angle Aand\angle B$ are acute angles such that tan A = tan B then prove that $\angle A=\angle B$.

Q.

If two towers of height $h_{1}and{h}_2$ subtend angles ${45}^{\circ }and{30}^{\circ }$ respectively at the midpoint of the line joining their feet, then find the ratio of

$h_1:h_2.$

1. $1:2$

2. $3:2$

3. $\sqrt{3}:1$

4. $1:\sqrt{3}$

Q.

Solution of the equation $3\tan (\theta –15°)=\tan (\theta +15°)$ is

1. $\theta =n\pi –\frac{\pi }{3}$

2. $\theta =n\pi +\frac{\pi }{3}$

3. $\theta =n\pi -\frac{\pi }{4}$

4. $\theta =n\pi +\frac{\pi }{4}$

Q. If $secA=\frac{2}{\sqrt{3}},$ find the value of $\frac{tanA}{cosA}+\frac{1+sinA}{tanA}$

Q.

If $\tan \left(x\right)=\frac{5}{12}$ in the third quadrant, then find the other trigonometric ratios.

Q.

State whether the following are true or false. Justify your answer.

(i) The value of tan A is always less than 1.

(ii) sec A =for some value of angle A.

(iii) cos A is the abbreviation used for the cosecant of angle A.

(iv) cot A is the product of cot and A

(v) sin θ =, for some angle θ