The sum of the series: sinp + x sin p + q + x 2 2 ! sin ( p + 2 q ) + ⋯ ∞ is

e x cos q [ sin ( p + x sin q ) ]

e xcos q [ cos ( p + xsin q ) ]

For non-negative integers n , let f n = ∑ k = 0 n sin ⁡ k + 1 n + 2 π sin ⁡ k + 2 n + 2 π ∑ k = 0 n sin 2 ⁡ k + 1 n + 2 π Assuming cos - 1 ⁡ x takes values in 0 , π , which of the following options is/are correct?

If α = tan ⁡ cos - 1 ⁡ f 6 , then α 2 + 2 α - 1 = 0

MEDIUM

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Find the value of $96 \cdot \cos \frac{π}{33} \cdot \cos \frac{2 π}{33} \cdot \cos \frac{4 π}{33} . . . . . . . . . \cos \frac{16 π}{33}$

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Important Questions on Trigonometric Ratios and Identities

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

96 \cos \frac{π}{33} \cos \frac{2 π}{33} \cos \frac{4 π}{33} \cos \frac{8 π}{33} \cos \frac{16 π}{33}

is equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value of

\cos \frac{π}{2^{2}} \cdot \cos \frac{π}{2^{3}} \cdot \dots \cdot \cos \frac{π}{2^{10}} \cdot \sin \frac{π}{2^{10}}

is:

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

e^{i θ} = cis θ

then

\sum_{n = 0}^{\infty} \frac{\cos (n θ)}{2^{n}} =

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

\lim_{n \to \infty} \frac{1}{n} [\sin \frac{π}{4} + \sin \frac{π}{12} (3 + \frac{1}{n}) + \sin \frac{π}{12} (3 + \frac{2}{n}) + . . . + \sin \frac{π}{3}] =

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The sum of the series:

sinp + x \sin (p + q) + \frac{x^{2}}{2!} \sin (p + 2 q) + \dots \infty

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value of

\cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15} \cos \frac{14 π}{15}

is equal to

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value of

2 \sin \frac{π}{22} \sin \frac{3 π}{22} \sin \frac{5 π}{22} \sin \frac{7 π}{22} \sin \frac{9 π}{22}

is equal to:

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

\sum_{k = 1}^{6} (\sin \frac{2 π k}{7} - i \cos \frac{2 π k}{7}) =

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

S = \sin^{2} p - \frac{1}{2} \sin (2 p) (\sin^{2} p) + \frac{1}{3} \sin (3 p) (\sin^{3} p) - \frac{1}{4} \sin (4 p) ((\sin^{4} p) + \dots \infty

then

\tan S =

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

In a triangle

A B C

, if

A : B : C = 1 : 2 : 4

, then

\sec A \sec B \sec C

is equal to

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

For non-negative integers

n,

let

f (n) = \frac{\sum_{k = 0}^{n} \sin (\frac{k + 1}{n + 2} π) \sin (\frac{k + 2}{n + 2} π)}{\sum_{k = 0}^{n} \sin^{2} (\frac{k + 1}{n + 2} π)}

Assuming

\cos^{- 1} x

takes values in

[0, π],

which of the following options is/are correct?

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

Represent the union of two sets by Venn diagram for each of the following.

$X = {x | x$ is a prime number between $80$ and $100}$

$Y = {y | y$ is an odd number between $90$ and $100}$

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value of

\cos \frac{4 π}{5} \cos \frac{6 π}{5} \cos \frac{8 π}{5}

is equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value of

\sin \frac{π}{n} + \sin \frac{3 π}{n} + \sin \frac{5 π}{n} + . . .

upto

n

terms is

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

If $16 \cos (\frac{2 π}{15}) \cos (\frac{4 π}{15}) \cos (\frac{8 π}{15}) \cos (\frac{16 π}{15}) = n$ , The value of $n$ is

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value of

\cos 20 ° + \cos 40 ° + . . . . . . + \cos 140 °

is not equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value of

\cos 20 ° + \cos 40 ° + . . . . . . + \cos 140 °

is not equal to

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

Let

p_{n} = c o s (\frac{2^{n} π}{65})

and

q_{n} = \cos (\frac{(4 n - 2) π}{65}),

where

n \in N .

S = (\frac{\sum_{n = 1}^{16} q_{n}}{\prod_{n = 1}^{5} p_{n}})

then find the value of

S .

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

The value(s) of

\cos \frac{π}{7} . \cos \frac{4 π}{7} . \cos \frac{5 π}{7}

is (are)

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Series

Given,

\cos 2^{m} θ \cos 2^{m + 1} θ \dots . . \cos 2^{n} θ = \frac{\sin 2^{n + 1} θ}{2^{n - m + 1} \sin 2^{m} θ}

, where

2^{m} θ \neq k π, n, m \in I,

then

(\cos 2^{3} \frac{π}{10} \cos 2^{4} \frac{π}{10} \cos 2^{5} \frac{π}{10} \dots . . \cos 2^{10} \frac{π}{10})

is equal to

Find the value of 96·cosπ33·cos2π33·cos4π33.........cos16π33

Important Questions on Trigonometric Ratios and Identities

Learn and Prepare for any exam you want !

Find the value of $96 \cdot \cos \frac{π}{33} \cdot \cos \frac{2 π}{33} \cdot \cos \frac{4 π}{33} . . . . . . . . . \cos \frac{16 π}{33}$