Solution: Option (A) $2 \sqrt{3}$ is correct. Explanation: As per the question, $\tan 75^{\circ}-\cot 75^{\circ}$ $\begin{array}{l} =\frac{\sin 75^{\circ}}{\cos 75^{\circ}}-\frac{\cos...
If and lies in third quadrant, then the value of is A. B. C. D.
Solution: Option (C) $-3 / \sqrt{10}$ is the correct. Explanation: As per the question, It is given that, $\tan \theta=3$ and $\theta$ lies in the third quadrant $\Rightarrow \cot \theta=1 / 3$ It...
The value of is A. B. C. D.
Solution: Option (B) 0 is correct. Explanation: As per the question, As $\cos 90^{\circ}=0$ We have, $\Rightarrow \cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \cos 90^{\circ} \ldots \cos...
The value of is A. B. C. D.
Solution: Option (C) $\sqrt{3 / 2}$ is correct. Explanation: As per the question, Let's say $\theta=15^{\circ} \Rightarrow 2 \theta=30^{\circ}$ Now, as it is known that, $\begin{array}{l}...
The value of is A. 0 B. 1 C. D.
Solution: Option (B) 1 is correct. Explanation: As per the question, $\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$ $=\tan 1^{\circ} \tan 2^{\circ} \ldots \tan 45^{\circ} \tan...
Which of the following is not correct? A. B. C. D.
Solution: Option (C) $\sec \theta=1 / 2$ Explanation: As per the question, It is known that, a) $\sin \theta=-1 / 5$ is correct since $\operatorname{Sin} \theta \in[-1,1]$ b) $\cos \theta=1$ is...
If and tan , then the value of is A. B. C. D.
Solution: Option (D) $\pi / 4$ is correct. Explanation: As per the question, $\tan \theta=\frac{1}{2}$ and $\tan \phi=\frac{1}{3}$ It is known that, $\begin{array}{l} \tan (\theta+\phi)=\frac{\tan...
If , then A. B. C. D. [Hint: A.M ≥ G.M.]
Solution: Option (D) $f(x) \geq 2$ is correcct. Explanation: As per the question, $f(x)=\cos ^{2} x+\sec ^{2} x$ It is known that, A.M $\geq$ G.M. $\begin{array}{l} \Rightarrow \frac{\cos ^{2}...
If , then is equal to A. 1 B.4 C. 2 D. None of these
Solution: Option (C) is the correct option. Explanation: As per the question, $\sin \theta+\operatorname{cosec} \theta=2$ Now square both the sides L.H.S. and R.H.S., We have, $\begin{array}{l}...
Find the general solution of the equation [Hint: Put which gives
Solution: Let's say, $r$ sina $=\sqrt{3}-1$ and $r \cos a=\sqrt{3}+1$ So, $\left.r=\sqrt{(}(\sqrt{3}-1)^{2}+(\sqrt{3}+1)^{2}\right\}=\sqrt{8}=2 \sqrt{2}$ And, tan $\alpha=(\sqrt{3}-1) /(\sqrt{3}+1)$...
Find the general solution of the equation
Solution: As per the question, $\sin x-3 \sin 2 x+\sin 3 x=\cos x-3 \cos 2 x+\cos 3 x$ Now grouping $\sin x$ and $\sin 3 x$ in Left Hand Side and, $\cos x$ and $\cos 3 x$ in Right Hand Side, We...
Find the general solution of the equation
Solution: As per the question, $5 \cos ^{2} \theta+7 \sin ^{2} \theta-6=0$ It is known that, $\sin ^{2} \theta=1-\cos ^{2} \theta$ So, $5 \cos ^{2} \theta+7\left(1-\cos ^{2} \theta\right)-6=0$...
If lies in the first quadrant and , then find the value of
Solution: As per the question, $\begin{array}{l} \cos \theta=8 / 17 \\ \left.\sin \theta=\pm \sqrt{(} 1-\cos ^{2} \theta\right) \end{array}$ As, $\theta$ lies in the first quadrant, so only positive...
If and , then show that [Hint: Find and then show
Solution: As per the question, $x=\sec \phi-\tan \phi$ and $y=\operatorname{cosec} \phi+\cot \phi$ It is given that, $\text {Left Hand Side}=x y+x-y+1$ $\begin{array}{l} =(\sec \phi-\tan...
If has and as its roots, then prove that [Hint: Use the identities and
Solution: As per the question, $a \cos 2 \theta+b \sin 2 \theta=c$ $\alpha$ and $\beta$ are the roots of the equation. Now using the multiple angles formula, It is known that, $\begin{array}{l} \cos...
Find the value of the expression
Solution: As per the question, Let's say, $y=3\left[\sin ^{4}(3 \pi / 2-a)+\sin ^{4}(3 \pi+\alpha)\right]-2\left[\sin ^{6}(\pi / 2+\alpha)+\sin ^{6}(5 \pi-\alpha)\right]$ It is known that,...
If cos (θ + ϕ) = m cos (θ – ϕ), then prove that tan θ = ((1 – m)/(1 + m)) cot ϕ [Hint: Express cos (θ + ϕ)/ cos (θ – ϕ) = m/l and apply Componendo and Dividendo]
Solution: As per the question, $\begin{array}{l} \cos (\theta+\phi)=\mathrm{m} \cos (\theta-\phi) \\ \because \cos (\theta+\phi)=\mathrm{m} \cos (\theta-\phi) \\ \Rightarrow \frac{\cos...
If and , then prove that
Solution: As per the question, $\sin (\theta+\alpha)=\mathrm{a}$ and $\sin (\theta+\beta)=\mathrm{b}$ $\text {Left Hand Side}=\cos 2(a-\beta)-4 a b \cos (\alpha-\beta)$ Now, using $\cos 2 x=2 \cos...
If , where , then find the value of
Solution: As per the question, $\sec x \cos 5 x=-1$ $\Rightarrow \cos 5 x=-1 / \sec x$ It is known that, $\sec x=1 / \cos x$ $\Rightarrow \cos 5 x+\cos x=0$ Now by using the transformation formula...
If , where , then find the value of .
Solution: As per the question, $2 \sin ^{2} \theta=3 \cos \theta$ It is known that, $\sin ^{2} \theta=1-\cos ^{2} \theta$ Provided that, $2 \sin ^{2} \theta=3 \cos \theta$ $2-2 \cos ^{2} \theta=3...
Find sin x/2, cos x/2 and tan x/2 in each of the following: sin x = 1/4, x in quadrant II
HERE,
Find sin x/2, cos x/2 and tan x/2 in each of the following: cos x = -1/3, x in quadrant III
HERE,
Find sin x/2, cos x/2 and tan x/2 in each of the following:
\[cos\text{ }x\text{ }=\text{ }-3/5\] FORMULA SUGGESTS,
Prove that:
HERE,
Prove that:
HERE,
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
HERE,
Prove that:
\[LHS\text{ }=\text{ }\left( cos\text{ }x\text{ }\text{ }cos\text{ }y \right)\text{ }2\text{ }+\text{ }\left( sin\text{ }x\text{ }\text{ }sin\text{ }y \right)\text{ }2\] By extending utilizing...
If , then find the general value of .
Solution: As per the question, $\Rightarrow \frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}=2 \operatorname{cosec} \theta$ As, $\sin ^{2} \theta+\cos ^{2} \theta=1$ $\Rightarrow...
Prove that:
Consider \[LHS\text{ }=\text{ }\left( cos\text{ }x\text{ }+\text{ }cos\text{ }y \right)\text{ }2\text{ }+\text{ }\left( sin\text{ }x\text{ }\text{ }sin\text{ }y \right)\text{ }2\] we get \[=\text{...
Find the most general value of satisfying the equation and
Solution: As per the question, We get, $\tan \theta=-1$ And $\cos \theta=1 / \sqrt{2}$ $\Rightarrow \theta=-\pi / 4$ Therefore, it is known that, $\theta$ lies in IV quadrant. $\theta=2 \pi-\pi /...
If , then find the general value of .
Solution: As per the question, $\sin \theta+\cos \theta=1$ Since, $\sin \theta+\cos \theta=1$ $\Rightarrow \sqrt{2}\left(\frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta\right)=1$ It is...
Prove that: (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0
Consider \[LHS\text{ }=\text{ }\left( sin\text{ }3x\text{ }+\text{ }sin\text{ }x \right)\text{ }sin\text{ }x\text{ }+\text{ }\left( cos\text{ }3x\text{ }\text{ }cos\text{ }x \right)\text{...
Prove that:
\[=\text{ }0\] = RHS
If , then show that [Hint: Express
Solution: It is known that, $\tan \theta=\frac{\sin \alpha-\cos \alpha}{\sin \alpha+\cos \alpha}$ $\Rightarrow \tan \theta=\frac{\cos \alpha\left(\frac{\sin \alpha}{\cos \alpha}-1\right)}{\cos...
Find the general solution for each of the following equation: sin x + sin 3x + sin 5x = 0
GIVEN: \[sin\text{ }x\text{ }+\text{ }sin\text{ }3x\text{ }+\text{ }sin\text{ }5x\text{ }=\text{ }0\] OR, \[\left( sin\text{ }x\text{ }+\text{ }sin\text{ }5x \right)\text{ }+\text{ }sin\text{...
Find the general solution for each of the following equation: sec2 2x = 1 – tan 2x
GIVEN: \[sec2\text{ }2x\text{ }=\text{ }1\text{ }\text{ }tan\text{ }2x\] OR, \[1\text{ }+\text{ }tan2\text{ }2x\text{ }=\text{ }1\text{ }\text{ }tan\text{ }2x\] \[tan2\text{ }2x\text{ }+\text{...
If , then show that [Hint: Use componendo and Dividendo]
Solution: As per the question, $\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}$ $\text { Since, } \sin (A+B)=\sin A \cos B+\cos A \sin B$ $\therefore \frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}$...
Find the general solution for each of the following equation: sin 2x + cos x = 0
GIVEN: \[sin\text{ }2x\text{ }+\text{ }cos\text{ }x\text{ }=\text{ }0\] OR, \[2\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }+\text{ }cos\text{ }x\text{ }=\text{ }0\] OR, \[cos\text{...
If , then prove that .
Solution: As per the question, $\cos \alpha+\cos \beta=0=\sin \alpha+\sin \beta \ldots \dots(i)$ As, $\text {Left Hand Side} =\cos 2 \alpha+\cos 2 \beta$ It is known that, $\cos 2 x=\cos ^{2} x-\sin...
Find the general solution for each of the following equation: cos 3x + cos x – cos 2x = 0
HERE,
Find the general solution for each of the following equation: cos 4x = cos 2x
HERE,
Find the principal and general solutions of the following equation: cosec x = – 2
HERE,
Find the principal and general solutions of the following equation: cot x = – √3
HERE,
Find the principal and general solutions of the following equation: sec x = 2
HERE,
Find the principal and general solutions of the following equation: tan x = √3
HERE,
If tan (A + B) = p, tan (A – B) = q, then show that tan 2A = (p + q) / (1 – pq). [Hint: Use 2A = (A + B) + (A – B)]
Solution: It is known that, $\tan 2 A=\tan (A+B+A-B)$ Also, $\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}$ $\therefore \tan 2 A=\frac{\tan (A+B)+\tan (A-B)}{1-\tan (A+B) \tan (A-B)}$ Now...
If and , then prove that [Hint: , then use
Solution: As per the question, $\tan \theta+\sin \theta=m \ldots \dots$...(i) $\tan \theta-\sin \theta=n \ldots \dots$ (ii) Now add eq.(i) and eq.(ii), $2 \tan \theta=m+n \ldots \dots$ (iii) Now...
Prove the following: cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1
ACCORDING TO THE QUES, \[L.H.S.\text{ }=\text{ }cos\text{ }6x\] OR, \[=\text{ }cos\text{ }3\left( 2x \right)\] FORMULA SAYS, \[~cos\text{ }3A\text{ }=\text{ }4\text{ }cos3\text{ }A\text{ }\text{...
Prove that
Solution: $\sin 4 A=\sin (2 A+2 A)$ It is known that, $\sin (A+B)=\sin A \cos B+\cos A \sin B$ As a result, $\sin 4 A=\sin 2 A \cos 2 A+\cos 2 A \sin 2 A$ $\Rightarrow \sin 4 A=2 \sin 2 A \cos 2 A$...
Prove the following: cos 4x = 1 – 8sin2 x cos2 x
ACCORDING TO THE QUES, \[LHS\text{ }=\text{ }cos\text{ }4x\] We can compose it as \[=\text{ }cos\text{ }2\left( 2x \right)\] Utilizing the equation \[cos\text{ }2A\text{ }=\text{ }1\text{ }\text{...
Prove the following:
ACCORDING TO THE QUES, \[LHS\text{ }=\text{ }tan\text{ }4x\text{ }=\text{ }tan\text{ }2\left( 2x \right)\] NOW,
Find the value of . [Hint: Let , use
Solution: Let's say, $\theta=45^{\circ}$ To find: $\tan 22^{\circ} 30^{\prime}=\tan (\theta / 2)$ It is known that, $\sin \theta=\cos \theta=1 / \sqrt{2}\left(\right.$ for...
Prove the following: cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES,
If a cos θ + b sin θ = m and a sin θ – b cos θ = n, then show that .
Solution: As per the question, $a \cos \theta+b \sin \theta=m \ldots \dots(i)$ $a \sin \theta-b \cos \theta=n \ldots \dots(ii)$ Now square and add eq.1 and eq.2, we have, $(a \cos \theta+b \sin...
Prove the following:
ACCORDING TO THE QUES, AND,
Prove that cos θ cos θ/2 – cos 3θ cos 9θ/2 = sin7θ sin4θ [Hint: Express L.H.S. = ½ [2cos θcos θ/2 – 2cos 3θ cos 9θ / 2]
Solution: By applying transformation formula, we have, $2 \cos A \cos B=\cos (A+B)+\cos (A-B)$ $-2 \sin A \sin B=\cos (A+B)-\cos (A-B)$ Now multiply and divide the expression by 2 . $\therefore...
If then find the value of
Solution: As per the question, $\tan x=b / a$ Let's say, $y=\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}$ $\therefore y=\sqrt{\frac{a\left(1+\frac{b}{a}\right)}{a\left(1-...
If and , where a lie between 0 and , find value of [Hint: Express tan as tan
Solution: As per the question, $\cos (\alpha+\beta)=4 / 5 \ldots \text {...(i) }$ It is known that, $\sin x=\sqrt{\left(1-\cos ^{2} x\right)}$ So, $\begin{array}{l} \sin...
Prove the following: cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
NOW, \[LHS\text{ }=\text{ }cot\text{ }4x~\left( sin\text{ }5x~+\text{ }sin\text{ }3x \right)\] OR ACCORDINGLY, \[=\text{ }2\text{ }cos\text{ }4x\text{ }cos\text{ }x\] LHS = RHS....
Prove the following: sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
ACCORDING TO THE QUES, By additional rearrangements \[=\text{ }2\text{ }sin\text{ }4x\text{ }cos\text{ }\left( \text{ }2x \right)\text{ }+\text{ }2\text{ }sin\text{ }4x\] OR \[=\text{ }2\text{...
If m sin θ = n sin (θ + 2α), then prove that tan (θ + α) cot α = (m + n)/(m – n) [Hints: Express sin(θ + 2α) / sinθ = m/n and apply componendo and dividend]
Solution: As per the question, $m \sin \theta=n \sin (\theta+2 a)$ We need to prove: $\tan (\theta+\alpha) \cot \alpha=(m+n) /(m-n)$ Proof: $\begin{array}{l} m \sin \theta=n \sin (\theta+2 a) \\...
Prove the following: cos2 2x – cos2 6x = sin 4x sin 8x
ACCORDING TO THE QUES, WE HAVE, \[=\text{ }\left[ 2\text{ }cos\text{ }4x\text{ }cos\text{ }\left( -\text{ }2x \right) \right]\text{ }\left[ -2\text{ }sin\text{ }4x\text{ }sin\text{ }\left( -\text{...
Prove the following: sin2 6x – sin2 4x = sin 2x sin 10x
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES, NOW SINCE WE KNOW,
Prove the following: sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
ACCORDING TO THE QUES, \[LHS\text{ }=\text{ }sin\text{ }\left( n~+\text{ }1 \right)x~sin\text{ }\left( n~+\text{ }2 \right)x~+\text{ }cos\text{ }\left( n~+\text{ }1 \right)x~cos\text{ }\left(...
Prove the following:
ACCORDING TO THE QUES, OR, \[=\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }\left( tan\text{ }x\text{ }+\text{ }cot\text{ }x \right)\] WE HAVE,
Prove the following:
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES,
Prove the following:
ACCORDING TO THE QUES,
Find the value of: (i) sin 75o (ii) tan 15o
\[\left( ii \right)\text{ }tan\text{ }15{}^\circ \] OR, \[=\text{ }tan\text{ }\left( 45{}^\circ \text{ }\text{ }30{}^\circ \right)\] Using formula
Prove that:
ACCORDING TO QUES,
Prove that:
ACCORDING TO QUES,
Prove that:
ACCORDING TO QUES, NOW, \[=\text{ }1/2\text{ }+\text{ }4/4\] \[=\text{ }1/2\text{ }+\text{ }1\] \[=\text{ }3/2\] \[=\text{ }RHS\]
Prove that:
ACCRDING TO QUES,
If [2sinα / (1+cosα+sinα)] = y, then prove that [(1– cosα+sinα) / (1+sinα)] is also equal to y. Hint: Express
Solution: As per the question, $y=2 \sin \alpha /(1+\cos \alpha+\sin \alpha)$ Now multiply the numerator and denominator by $(1-\cos \alpha+\sin \alpha)$, We have, $\Rightarrow y=\frac{2 \sin...
Prove that
Solution: As per the question, $\text { Left Hand Side }=\frac{\ tan A+\sec A-1}{\ tan A-\sec A+1}$ $=\frac{\frac{\sin A}{\cos A}+\frac{1}{\cos A}-1}{} {\frac{\sin A}{\cos A}-\frac{1}{\ cos A}+1}$...
Find the values of the trigonometric functions in
We realize that upsides of tan x repeat after a time frame of \[\pi \text{ }or\text{ }180{}^\circ \] So we get By additional computation
Find the values of the trigonometric functions in
We realize that upsides of transgression x repeat after a time frame of \[2\pi \text{ }or\text{ }360{}^\circ \] So we get By additional estimation
Find the values of the trigonometric functions in
We realize that upsides of tan x repeat after a timespan of \[\pi \text{ }or\text{ }180{}^\circ \] So we get By additional estimation We get \[=\text{ }tan\text{ }60o\] \[=\text{ }\surd...
Find the values of the trigonometric functions in cosec (–1410°)
We realize that upsides of cosec x repeat after a time frame of \[2\pi \text{ }or\text{ }360{}^\circ \] So we get By additional computation \[=\text{ }cosec\text{ }30o\text{ }=\text{ }2\]...
Find the values of the trigonometric functions sin 765°
We realize that upsides of sin x repeat after a time frame of $$ \[2\pi \text{ }or\text{ }360{}^\circ \] So we get By additional computation \[=\text{ }sin\text{ }45o\] \[=\text{ }1/\surd...
Find the values of other five trigonometric functions tan x = -5/12, x lies in second quadrant
It is given that \[tan\text{ }x\text{ }=\text{ }\text{ }5/12\] We can compose it as We realize that \[1\text{ }+\text{ }tan2\text{ }x\text{ }=\text{ }sec2\text{ }x\] We can compose it as...
Find the values of other five trigonometric functions sec x = 13/5, x lies in fourth quadrant
It is given that sec x = 13/5 We can write it as We know that sin2 x + cos2 x = 1 We can write it as sin2 x = 1 – cos2 x Substituting the values sin2 x = 1 – (5/13)2 sin2 x = 1 – 25/169 = 144/169...
cot x = 3/4, x lies in third quadrant
It is given that We can compose it as We realize that We can compose it as Subbing the qualities Here x lies in the third quadrant so the worth of sec x will be negative We can compose it...
Find the values of other five trigonometric functions sin x = 3/5, x lies in second quadrant
It is given that sin x = 3/5 We can write it as We know that sin2 x + cos2 x = 1 We can write it as cos2 x = 1 – sin2 x
Find the values of other five trigonometric functions cos x = -1/2, x lies in third quadrant.
according to ques,
Find the angle in radian though which a pendulum swings if its length is 75 cm and the tip describes an arc of length (i) 10 cm (ii) 15 cm
Surrounded by sweep r unit, if a circular segment of length l unit subtends a point θ radian at the middle, then, at that point, \[\theta \text{ }=\text{ }1/r\] We realize that \[r\text{ }=\text{...
In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
The components of the circle are Width = \[40\text{ }cm\] Range = \[40/2\text{ }=\text{ }20\text{ }cm\] Consider AB be as the harmony of the circle for example length = \[20\text{ }cm\] In ΔOAB,...
Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm (Use π = 22/7)
Find the degree measures corresponding to the following radian measures (Use π = 22/7) (i) 11/16 (ii) -4
\[\left( i \right)\]$$ \[11/16\] Here \[\pi \text{ }radian\text{ }=\text{ }180{}^\circ \] \[\left( ii \right)\text{ }-4\] Here \[\pi \text{ }radian\text{ }=\text{ }180{}^\circ \]
Find the radian measures corresponding to the following degree measures: (iii)
(iv)
(iv) $$ \[520{}^\circ \]
Find the radian measures corresponding to the following degree measures: (i)
(ii)
is the answer for this. is the answer for this