Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

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If $f (x) = \cos (4 π {x}) + \sin (4 π {x})$ where $\{x\}$ denotes fractional part of $x$ then which of the following is true?

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Important Questions on Trigonometric Ratios and Identities

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Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If

f (x) = \sqrt{[(\sin 2 x + 2) (\cos 2 x + 2)]}

for

x

is a real number, then which of the following values lies in the domain of

f (x)

but NOT in the range of

f (x) ?

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The number of solutions of the equation

|\cot x| = \cot x + \frac{1}{\sin x}, 0 \leq x \leq 2 π

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The equation

\sin (e^{x}) = 5^{x} + 5^{- x}

has exactly how many solutions?

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of

\tan 1 ° \tan 2 ° \tan 3 ° \dots \tan 89 °

is

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If

A = \{x \in [0, 2 π] / \tan x - \tan^{2} x > 0\}

and

B = \{x \in [0, 2 π] / | \sin x | < \frac{1}{2}\},

then

A \cap B =

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of

\cos 1200 ° + \tan 1485 °

is

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If

x = \cos 1 ° \cos 2 ° \cos 3 ° \dots \cos 89 °

and

y = \cos 2 ° \cos 6 ° \cos 10 ° \dots \cos 86 °,

then what is the integer nearest to

\frac{2}{7} \log_{2} (\frac{y}{x}) ?

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

Let

θ

be an angle in the standard position such that the point

(- 5, 12)

lies on its terminal side, then

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The graphs of the polynomial

x^{2} - 1

and

\cos x

intersect

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If

\cos θ = - \frac{\sqrt{3}}{2}

and

\sin α = - \frac{3}{5}

, where

θ

does not lie in the third quadrant, then

\frac{2 \tan α + \sqrt{3} \tan θ}{\cot^{2} θ + \cos α}

is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The range of the function

f (x) = \log_{\sqrt{5}} (3 + \cos (\frac{3 π}{4} + x) + \cos (\frac{π}{4} + x) + \cos (\frac{π}{4} - x) - \cos (\frac{3 π}{4} - x))

is :

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

\tan 1 ° \times \tan 2 ° \times \tan 3 ° \times \dots \dots \dots \dots \times \tan 89 ° =

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of

2 \sin (\frac{π}{8}) \sin (\frac{2 π}{8}) \sin (\frac{3 π}{8}) \sin (\frac{5 π}{8}) \sin (\frac{6 π}{8}) \sin (\frac{7 π}{8})

is :

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If

W, X, Y

and

Z

are four angles taken in order of a cyclic quadrilateral, then the value of

(13 + \cot W + \cot X + \cot Y + \cot Z + 13)

is

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

\cot \frac{π}{16} \cdot \cot \frac{2 π}{16} \cdot \cot \frac{3 π}{16} \cdot \cot \frac{4 π}{16} \cdot \cot \frac{5 π}{16} \cdot \cot \frac{6 π}{16} \cdot \cot \frac{7 π}{16} =

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The maximum and minimum value of function

f (x) = \sin 3 x + 4

are respectively :

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

{(\frac{\sin 35 °}{\cos 55 °})}^{2} + {(\frac{\cos 55 °}{\sin 35 °})}^{2} - 2 \cos 30 ° =

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

The value of

\frac{\tan 330 ° \sec 420 ° \sin 300 °}{\tan 135 ° \sin 210 ° \sec 315 °}

is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If

\sec θ = \frac{13}{12}, θ

lies in

4^{th}

quadrant, then

\tan θ \times cosec θ \times \sin θ \times \cos θ =

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Functions and Properties

If

\cos (θ_{1}) + \cos (θ_{2}) + \cos (θ_{3}) + \cos (θ_{4}) = - 4,

then the value of

\cot (\frac{θ_{1}}{2}) + \cot (\frac{θ_{2}}{2}) + \cot (\frac{θ_{3}}{2}) + \cot (\frac{θ_{4}}{2}) =