Relationship between Trigonometric Ratios

Q. Express the ratios $cosA$ and $sinA$ respectively in terms of $tanA$.

1. $\frac{tanA}{\sqrt{1+ta{n}^{2}A}},\frac{1}{\sqrt{1+ta{n}^{2}A}}$
2. $\frac{1}{\sqrt{1+ta{n}^{2}A}},\frac{tanA}{\sqrt{1+ta{n}^{2}A}}$
3. $\frac{tanA}{\sqrt{1-ta{n}^{2}A}},\frac{1}{\sqrt{1-ta{n}^{2}A}}$
4. $\frac{1}{\sqrt{1-ta{n}^{2}A}},\frac{tanA}{\sqrt{1-ta{n}^{2}A}}$

Q.

In a $∆ABC$, if the sides are $a=3,b=5$ and $c=4$, then $\sin \left(\frac{B}{2}\right)+\cos \left(\frac{B}{2}\right)$ is equal to

1. $\sqrt{2}$

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

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

4. $1$

Q.

The value of tan 45 is

1. 1

2. 0

3. 90

4. 45

Q.

Prove: $\sin 3a=3\sin a-4{\sin }^{3}a$, when $a={30}^{\circ }$

Q.

Find the values of $cot\theta andcosec\theta$ in the figure given above.

1. $2,\frac{5}{2}$

2. $\frac{4}{3},\frac{5}{3}$

3. $2,\frac{5}{4}$

4. $\frac{5}{2},2$

Q.

If $P,Q\text{and}R$are the interior angles of trinagles $PQR$, then show that $\tan \left(\frac{\mathrm{Q}+\mathrm{R}}{2}\right)=\cot \left(\frac{\mathrm{P}}{2}\right)$.

Q.

The value of $\frac{tan{63}^{\circ }+cot{23}^{\circ }}{tan{27}^{\circ }+cot{67}^{\circ }}-tan{63}^{\circ }tan{67}^{\circ }$ is

1. 1

2. 0

3. -1

4. 2

Q.

Question 8
If cot A = $\frac{4}{3}$, check whether $\frac{(1-ta{n}^{2}A)}{(1+ta{n}^{2}A)}=co{s}^{2}A-si{n}^{2}A$ or not.

Q.

Find the Length of all sides of the given right angled triangle if the area is 36 $c{m}^2$

1. AB = 6cm , BC = 12cm, CA = 18

2. AB = 12cm , BC = 6cm, CA = 6$\sqrt{3}$

3. AB = 12cm , BC = 6cm, CA = 3$\sqrt{6}$

4. AB = 6cm , BC = 12cm, CA = 6$\sqrt{3}$

Q.

State True/False

In any $\Delta ABC$ where $\angle B={90}^{\circ },si{n}^{2}A+co{s}^{2}C=1$

1. False

2. True

Q.

If sin4A = cos(A - 30), where 4A and (A - 30) are acute angles, then the value of A is ___.

Q. If $4\tan$$\theta$$=3$, then evaluate $\frac{(4\sin \theta -\cos \theta )}{(4\sin \theta +\cos \theta )}$

Q.

If sin $\alpha$ = $\frac{5}{13}$ and sin $\beta$ = $\frac{4}{5}$ then the value of cos $\alpha$ cos $\beta$ - sin $\alpha$ sin $\beta$ is

1. $\frac{16}{65}$

2. $\frac{9}{65}$

3. $\frac{11}{65}$

4. $\frac{15}{65}$

Q. If $2\cos 2A=\sqrt{3}$ and A is acute, find $\sin 3A$.

1. $\frac{3}{\sqrt{2}}$
2. $\frac{1}{\sqrt{2}}$
3. $\frac{1}{2}$
4. $0$

Q. If $\tan \theta +\cot \theta =5$, then ${\tan }^2\theta +{\cot }^2\theta =$

1. $27$
2. $29$
3. $24$
4. $23$

Q. What is the value of $\frac{si{n}^2{63}^{\circ }+si{n}^2{27}^{\circ }}{co{s}^2{17}^{\circ }+co{s}^2{73}^{\circ }}$?

Q. On simplifying the expression $(\sec A+\tan A)(1-\sin A)$, we get

1. 1 + sin A
2. 1 + cos A
3. cos A
4. sin A

Q.

There are 3 squares which are arranged in such a manner that it forms a triangle as shown in the figure below. If the area of square A and B is $81c{m}^2$ and $144c{m}^2$ respectively, then find the area of the third square C.

1. 

2. 

3. 

4. 

Q. If 2 cos 2A= root 3 and A is acute
1. Sin square ( 75 - A ) + cos square ( 45 +A )

Q.

Match the following:

$$\begin{array}{cc}1.cos({90}^{\circ }-A)& a)secA\\ 2.cot({90}^{\circ }-A)& b)tanA\\ 3.cosec({90}^{\circ }-A)& c)sinA\end{array}$$

1. 1 - a; 2 - b; 3 - c

2. 1- c; 2 - b; 3 - a

3. 1 - a; 2 - c; 3 - b

4. 1 - b; 2 - c;3 - a

Q. A=B=45_​​​​ 1) SIN (A-B) =SIN A cos B - cos A sin B

Q. If $sec\theta =\frac{5}{4},$ find the value of $\frac{sin\theta -2cos\theta }{tan\theta -cot\theta }$

Q. $\Delta ABC$ is right angled at B, given $5\times sinA=3$. Find cos($\angle$C) + tan($\angle$A) + cosec($\angle$C).

1. $\frac{12}{5}$
2. $\frac{13}{12}$
3. $\frac{13}{5}$
4. $\frac{12}{5}$

Q.

1 + $ta{n}^{2}A$ = $se{c}^{2}A$ is valid for which of the following ranges

1. 0 A 90

2. 0 A 90

3. 0 A 90

4. 0 A 90

Q. $\Delta ABC$ is right angled at B, given $5\times sinA=3$.
Find cos($\angle$C)+tan($\angle$A) + cosec($\angle$C).

1. $\frac{12}{5}$
2. $\frac{13}{12}$
3. $\frac{13}{5}$
4. $\frac{12}{5}$

Q. If $\tan \theta +\cot \theta =5,$ find the value of ${\tan }^2\theta +{\cot }^2\theta$.

Q.

$\text{If}17cos\theta =15$,
$\text{then,}tan\theta +2sec\theta =$_____.

1. 2

2. 3

3. $2\frac{1}{5}$

4. $2\frac{4}{5}$

Q. If $\tan \theta =4$, where $\theta$ is an acute angle, find $\sec \theta$ using the identity.

Q. Evaluate $\frac{cos{70}^{\circ }}{sin{20}^{\circ }}+\frac{cos{59}^{\circ }}{sin{31}^{\circ }}-8si{n}^2{30}^{\circ }=0$

Q.

Match the following
$\begin{array}{cccc}& ListI& & ListII\\ A.& \frac{1+cosA-sinA}{1+cosA+sinA}& i.& \frac{1}{cosecA}+\frac{1}{secA}\\ B.& (1+\frac{1}{ta{n}^{2}A})(1+\frac{1}{co{t}^{2}A})& II.& secA-tanA\\ C.& \frac{cosA}{1-tanA}+\frac{sinA}{1-cotA}& iii.& \frac{1}{co{s}^{2}A-cos4A}\end{array}$

1. A-ii, B-iii, C-i

2. A-iii, B- ii, C-i

3. None of these

4. A- i, B-ii, C-iii