EASY

Earn 100

[iii] Prove that $\sqrt{3}\left(\tan 170^{\circ}-\tan 140^{\circ}\right)=1+\tan 170^{\circ} \tan 140^{\circ}$.

Important Questions on Trigonometric Ratios and Identities

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

\frac{\sqrt{3}}{\sin (20^{o})} - \frac{1}{\cos (20^{o})} =

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

x \sin^{3} θ + y \cos^{3} θ = \sin θ \cos θ

and

x \sin θ = y \cos θ

, then

x^{2} + y^{2}

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

\frac{\sin A - \sin B}{\cos A + \cos B}

is equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

The number of real numbers

λ

for which the equality

\frac{\sin (λ α)}{\sin α} - \frac{\cos (λ α)}{\cos α} = λ - 1,

holds for all real

α

which are not integral multiples of

\frac{π}{2}

is-

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

A B C D

is a trapezium such that

A B

and

C D

are parallel and

B C ⊥ C D

. If

∠ A D B = θ

B C = p

and

C D = q

, then

A B

is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

\sin θ - \cos θ = 1

,

then the value of

\sin^{3} θ - \cos^{3} θ

is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

If an angle

A

of a

Δ A B C

satisfies

5 \cos A + 3 = 0

, then the roots of the quadratic equation

9 x^{2} + 27 x + 20 = 0

are

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

Let

P = \{θ : \sin θ - \cos θ = \sqrt{2} \cos θ\}

and

Q = \{θ : \sin θ + \cos θ = \sqrt{2} \sin θ\},

be two sets. Then

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

\cos α + \cos β = \frac{3}{2} and \sin α + \sin β = \frac{1}{2}

and

θ

is the arithmetic mean of

α & β,

then

\sin 2 θ + \cos 2 θ

is equal to:

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

In a

Δ A B C

\frac{a}{b} = 2 + \sqrt{3}

, and

∠ C = 60 ° .

Then the ordered pair

(∠ A, ∠ B)

is equal to:

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

Let

f_{k} (x) = \frac{1}{k} (\sin^{k} x + \cos^{k} x)

where

x \in R

and

k \geq1

. Then

f_{4} (x) - f_{6} (x)

equals

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

s i n^{4} α + 4 c o s^{4} β + 2 = 4 \sqrt{2} s i n α c o s β

α, β \in [0, π]

, then

c o s (α + β) - c o s (α - β)

is equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

cosec θ = \frac{p + q}{p - q} (p \neq q, p \neq 0),

then

|\cot (\frac{π}{4} + \frac{θ}{2})|

is equals to:

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

\sin 15^{o} =

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

A > 0, B > 0

and

A + B = \frac{π}{6}

, then the minimum positive value of

(\tan A + \tan B)

is :

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

The value of

\sec^{- 1} (\frac{1}{4} \sum_{k = 0}^{10} \sec (\frac{7 π}{12} + \frac{k π}{2}) \sec (\frac{7 π}{12} + \frac{(k + 1) π}{2}))

in the interval

[- \frac{π}{4}, \frac{3 π}{4}]

equals

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

The angle of elevation of the top of a vertical tower from a point A, due east of it is

45^{o}

. The angle of elevation of the top of the same tower from a point B, due south of A is

30^{o}

. If the distance between A and B is

54 \sqrt{2} m

, then the height of the tower (in meters), is:

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

The integer part of the number

\sum_{k = 0}^{44} \frac{1}{cosk ° \cos (k + 1) °}

(\sin 1 ° = 0.0174524)

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

The expression

\frac{tanA}{1 - cotA} + \frac{cotA}{1 - tanA}

can be written as :

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Trigonometric Identities

The value of

\sum_{k = 1}^{13} \frac{1}{\sin (\frac{π}{4} + \frac{(k - 1) π}{6}) \sin (\frac{π}{4} + \frac{k π}{6})}

is equal to

[iii] Prove that $\sqrt{3}\left(\tan 170^{\circ}-\tan 140^{\circ}\right)=1+\tan 170^{\circ} \tan 140^{\circ}$.

Important Questions on Trigonometric Ratios and Identities

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