MEDIUM

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If $\frac{\sin^{- 1} x}{a} = \frac{\cos^{- 1} x}{b} = \frac{\tan^{- 1} y}{c}; 0 < x < 1$ , then the value of $\cos (\frac{π c}{a + b})$ is

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Important Questions on Trigonometry

EASY

Using the information from the figure, $\cos^{- 1} x$ is equal to

Question Image

EASY

The value of

\cos (\sin^{- 1} \frac{π}{3} + \cos^{- 1} \frac{π}{3})

MEDIUM

θ = \sin^{- 1} x + \cos^{- 1} x - \tan^{- 1} x, x \geq 0

then the smallest interval in which

θ

lies is

EASY

The value of

\sin^{- 1} (- 1 / 2) + 60 °

HARD

\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2},

then the value of

x^{100} + y^{100} + z^{100} - \frac{9}{x^{101} + y^{101} + z^{101}}

is equal to

MEDIUM

\cos (2 \tan^{- 1} x) = \frac{1}{2},

then value of

x

HARD

Match the items of List-I with those of the items of List-II

	List- $I$		List- $II$
A	Range of $\sec^{- 1} [1 + \cos^{2} x]$ , $[\cdot]$ denote greatest integer function	I	odd function
B	Domain of $f (x)$ where $f (x + \frac{1}{x}) = x^{2} + \frac{1}{x^{2}}$	II	$\{0, \frac{1}{2}\}$
C	$f (x + y) = f (x) + f (y); f (1) = 5$	III	$\{\sec^{- 1} 5, \sec^{- 1} 4\}$
D	$\sin^{- 1} x - \cos^{- 1} x + \sin^{- 1} (1 - x) = 0 \Rightarrow x \in$	IV	$R$
		V	$\{\sec^{- 1} 1, \sec^{- 1} 2\}$

EASY

The principal value of

\tan^{- 1} (\tan \frac{3 π}{5})

HARD

\sin^{- 1} x + \sin^{- 1} y + \sin^{- 1} z = \frac{3 π}{2}

and

f (1) = 2, f (p + q) = f (p) \cdot f (q), \forall p, q \in R

, then

x^{f (1)} + y^{f (2)} + z^{f (3)} - \frac{(x + y + z)}{x^{f (1)} + y^{f (2)} + z^{f (3)}}

is equal to

HARD

Let $Z$ denote the set of integers. Then match the items in list- $I$ , with those of the items in List- $II$

	List- $I$		List- $II$
$(A)$	$\sin^{- 1} (\frac{2 \sqrt{2}}{3}) + \sin^{- 1} \frac{1}{3}$	$(I)$	$k π \pm {(- 1)}^{k} \frac{π}{6}, k \in Z$
$(B)$	$\sin^{- 1} (\frac{{(- 1)}^{n}}{2}), n \in Z$	$(II)$	$k π \pm 1, k \in Z$
$(C)$	$\tan^{- 1} (\sec \frac{π}{4} + \tan \frac{π}{4})$	$(III)$	$\frac{3}{2}$
$(D)$	$\sin^{- 1} \|\sin x\| = \sqrt{\sin^{- 1} \|\sin x\|} \Rightarrow x \in$	$(IV)$	$\frac{3 π}{8}$
		$(V)$	$\frac{π}{2}$