Trigonometric Functions of Compound Angles

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JEE Main 2010

Let $cos (α + β) = 4/5$ and $sin (α - β) = 5/13$ , where $0 \leq α, β \leq π /4$ . Then $tan 2 α =$

56/33

(B)

19/12

(C)

20/7

(D)

25/16

JEE Main 2006

LEVELJEE Main

If the roots of the quadratic equation $x^{2} + p x + q = 0$ are $tan 3 0^{\circ}$ and $tan 1 5^{\circ}$ , respectively, then the value of $2 + q - p$ is

In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

(A)

a = b + c

(B)

c = a + b

(C)

b = c

(D)

b = a + c

JEE Advanced 1988

LEVELBoard

The value of the expression $3 csc 2 0^{\circ} - sec 2 0^{\circ}$ is equal to

(A)

2

(B)

2 sin 2 0^{\circ} / sin 4 0^{\circ}

(C)

4

(D)

4 sin 2 0^{\circ} / sin 4 0^{\circ}

JEE Advanced 1983

LEVELJEE Main

If $f (x) = cos (ln x)$ , then $f (x) f (y) - \frac{1}{2} [f (x / y) + f (x y)]$ has the value

(A)

- 1

(B)

1/2

(C)

- 2

Determine the smallest positive value of $x$ (in degrees) for which $tan (x + 10 0^{\circ}) = tan (x + 5 0^{\circ}) tan x tan (x - 5 0^{\circ})$ .

JEE Advanced 1978

LEVELBoard

If $tan α = \frac{m}{m + 1}$ and $tan β = \frac{1}{2 m + 1}$ , find the possible values of $(α + β)$ .

JEE Advanced 1979

LEVELJEE Main

(a) Draw the graph of $y = \frac{1}{2} (sin x + cos x)$ from $x = - π /2$ to $x = π /2$ . (b) If $cos (α + β) = 4/5, sin (α - β) = 5/13$ , and $α, β$ lies between 0 and $π /4$ , find $tan 2 α$ .

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

If $cos (α + β) = \frac{3}{5}, sin (α - β) = \frac{5}{13}$ and $0 < α, β < \frac{π}{4}$ , then $tan (2 α)$ is equal to :

(A)

\frac{21}{16}

(B)

\frac{63}{52}

(C)

\frac{33}{52}

(D)

\frac{63}{16}

JEE Main 2022 (26 June Shift 2)

LEVELBoard

$16 sin (2 0^{\circ}) sin (4 0^{\circ}) sin (8 0^{\circ})$ is equal to :

(A)

3

(B)

23

(C)

(D)

43

JEE Main 2022 (28 June Shift 2)

LEVELJEE Main

If $cot α = 1$ and $sec β = - \frac{5}{3}$ , where $π < α < \frac{3 π}{2}$ and $\frac{π}{2} < β < π$ , then the value of $tan (α + β)$ and the quadrant in which $α + β$ lies, respectively are

(A)

- \frac{1}{7}

and

I V^{t h}

quadrant

(B)

7 and

I^{s t}

quadrant

(C)

-7 and

I V^{t h}

quadrant

(D)

\frac{1}{7}

and

I^{s t}

quadrant

JEE Main 2024 (01 Feb Shift 1)

LEVELJEE Main

If $tan A = \frac{1}{x ( x ^{2} + x + 1 )}$ , $tan B = \frac{x}{x ^{2} + x + 1}$ and $tan C = (x^{- 3} + x^{- 2} + x^{- 1})^{\frac{1}{2}}$ , $0 < A, B, C < \frac{π}{2}$ , then $A + B$ is equal to :

(A)

C

(B)

π - C

(C)

2 π - C

(D)

\frac{π}{2} - C

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let cos(α+β)=4/5 and sin(α−β)=5/13, where 0≤α,β≤π/4. Then tan2α=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the roots of the quadratic equation x2+px+q=0 are tan30∘ and tan15∘, respectively, then the value of 2+q−p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle PQR,∠R=π/2. If tan(P/2) and tan(Q/2) are the roots of ax2+bx+c=0,a=0 then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of the expression 3​csc20∘−sec20∘ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=cos(lnx), then f(x)f(y)−21​[f(x/y)+f(xy)] has the value

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Determine the smallest positive value of x (in degrees) for which tan(x+100∘)=tan(x+50∘)tanxtan(x−50∘).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If tanα=m+1m​ and tanβ=2m+11​, find the possible values of (α+β).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (a) Draw the graph of y=2​1​(sinx+cosx) from x=−π/2 to x=π/2. (b) If cos(α+β)=4/5,sin(α−β)=5/13, and α,β lies between 0 and π/4, find tan2α.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If cos(α+β)=53​,sin(α−β)=135​ and 0<α,β<4π​, then tan(2α) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 16sin(20∘)sin(40∘)sin(80∘) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If cotα=1 and secβ=−35​, where π<α<23π​ and 2π​<β<π, then the value of tan(α+β) and the quadrant in which α+β lies, respectively are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If tanA=x(x2+x+1)​1​, tanB=x2+x+1​x​​ and tanC=(x−3+x−2+x−1)21​, 0<A,B,C<2π​, then A+B is equal to :

Let $cos (α + β) = 4/5$ and $sin (α - β) = 5/13$ , where $0 \leq α, β \leq π /4$ . Then $tan 2 α =$

If the roots of the quadratic equation $x^{2} + p x + q = 0$ are $tan 3 0^{\circ}$ and $tan 1 5^{\circ}$ , respectively, then the value of $2 + q - p$ is

In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

The value of the expression $3 csc 2 0^{\circ} - sec 2 0^{\circ}$ is equal to

If $f (x) = cos (ln x)$ , then $f (x) f (y) - \frac{1}{2} [f (x / y) + f (x y)]$ has the value

Determine the smallest positive value of $x$ (in degrees) for which $tan (x + 10 0^{\circ}) = tan (x + 5 0^{\circ}) tan x tan (x - 5 0^{\circ})$ .

If $tan α = \frac{m}{m + 1}$ and $tan β = \frac{1}{2 m + 1}$ , find the possible values of $(α + β)$ .

(a) Draw the graph of $y = \frac{1}{2} (sin x + cos x)$ from $x = - π /2$ to $x = π /2$ . (b) If $cos (α + β) = 4/5, sin (α - β) = 5/13$ , and $α, β$ lies between 0 and $π /4$ , find $tan 2 α$ .

If $cos (α + β) = \frac{3}{5}, sin (α - β) = \frac{5}{13}$ and $0 < α, β < \frac{π}{4}$ , then $tan (2 α)$ is equal to :

$16 sin (2 0^{\circ}) sin (4 0^{\circ}) sin (8 0^{\circ})$ is equal to :

If $cot α = 1$ and $sec β = - \frac{5}{3}$ , where $π < α < \frac{3 π}{2}$ and $\frac{π}{2} < β < π$ , then the value of $tan (α + β)$ and the quadrant in which $α + β$ lies, respectively are

If $tan A = \frac{1}{x ( x ^{2} + x + 1 )}$ , $tan B = \frac{x}{x ^{2} + x + 1}$ and $tan C = (x^{- 3} + x^{- 2} + x^{- 1})^{\frac{1}{2}}$ , $0 < A, B, C < \frac{π}{2}$ , then $A + B$ is equal to :