Prove that: 3 sin - 1 x = sin - 1 3 x - 4 x 3 , x ∈ - 1 2 , 1 2 .

To prove: 3 sin - 1 x = sin - 1 3 x - 4 x 3 , x ∈ - 1 2 , 1 2 Let sin - 1 x = θ ∴ x = sin θ Now, consider, RHS = sin - 1 ( 3 x - 4 x 3 ) = sin - 1 ( 3 sin θ - 4 sin 3 θ ) = sin - 1 ( sin 3 θ ) [Using: sin 3 θ = 3 sin θ - 4 sin 3 θ ] = 3 θ = 3 sin - 1 x = LHS Hence, proved.

Find the value of tan - 1 3 - cot - 1 ( - 3 ) .

We have, tan - 1 3 - cot - 1 ( - 3 ) = tan - 1 3 - π - cot - 1 ( 3 ) = tan - 1 tan π 3 - π - cot - 1 cot π 6 = π 3 - π - π 6 = π 3 - 5 π 6 = 2 π - 5 π 6 = - 3 π 6 = - π 2 Hence, the value of tan - 1 3 - cot - 1 ( - 3 ) is - π 2 .

Prove that cos - 1 x = 2 cos - 1 1 + x 2

To prove: cos - 1 x = 2 cos - 1 1 + x 2 Let x = cos 2 θ cos - 1 cos 2 θ = 2 cos - 1 1 + cos 2 θ 2 = 2 cos - 1 2 cos 2 θ 2 = 2 cos - 1 cos θ = 2 θ

Prove that: sin - 1 2 x 1 - x 2 = 2 cos - 1 x , 1 2 ≤ x ≤ 1 .

To prove: sin - 1 2 x 1 - x 2 = 2 cos - 1 x , 1 2 ≤ x ≤ 1 Put x = cos θ ⇔ θ = cos - 1 x Now, consider, LHS = sin - 1 2 x 1 - x 2 = sin - 1 2 cos θ 1 - cos 2 θ = sin - 1 ( 2 cos θ sin θ ) = sin - 1 ( sin 2 θ ) [Using: sin 2 θ = 2 sin θ cos θ ] = 2 θ = 2 cos - 1 x = RHS Hence, proved.

Show that sin - 1 3 5 - sin - 1 8 17 = cos - 1 84 85

Let sin − 1 3 5 = x and sin − 1 8 17 = y Therefore sin x = 3 5 and siny = 8 17 Now cos x = 1 − sin 2 x = 1 − 9 25 = 4 5 and cos y = 1 − sin 2 y = 1 − 64 289 = 15 17 We have cos ( x - y ) = cosxcosy + sinxsiny = 4 5 × 15 17 + 3 5 × 8 17 = 84 85 Therefore x − y = cos − 1 84 85 Hence sin − 1 3 5 − sin − 1 8 17 = cos − 1 84 85

Prove that: sin - 1 3 5 - sin - 1 8 17 = c o s - 1 84 75

L.H.S. = sin - 1 3 5 - sin - 1 8 17 = sin - 1 3 5 1 - 8 17 2 - 8 17 1 - 3 5 2 = sin - 1 3 5 1 - 64 289 - 8 17 1 - 9 25 = sin - 1 3 5 225 289 - 8 17 16 25 = sin - 1 3 5 × 15 17 - 8 17 × 4 5 = sin - 1 45 - 32 85 = sin - 1 13 85 Let sin - 1 13 85 = A ⇒ A = sin 13 85 , A ∈ 0 , π 2 ∵ sin A > 0 ∴ cos A = 1 - sin 2 A = 1 - 13 85 2 = 7225 - 169 7225 = 84 85 ∵ cos A > 0 ∀ A ∈ 0 , π 2 ⇒ A = cos - 1 84 85 ∴ sin - 1 3 5 - sin - 1 8 17 = cos - 1 84 85 . Hence proved.

Solve: tan - 1 4 x + tan - 1 6 x = π 4 .

Given tan - 1 4 x + tan - 1 6 x = π 4 ⇒ tan - 1 4 x + 6 x 1 - 4 x · 6 x = π 4 (Using tan - 1 x + tan - 1 y = tan - 1 x + y 1 - x y , where x y < 1 ) ⇒ tan - 1 10 x 1 - 24 x 2 = π 4 ⇒ 10 x 1 - 24 x 2 = 1 ⇒ 10 x = 1 - 24 x 2 ⇒ 24 x 2 + 10 x - 1 = 0 ⇒ 24 x 2 + 12 x - 2 x - 1 = 0 ⇒ 12 x ( 2 x + 1 ) - 1 ( 2 x + 1 ) = 0 ⇒ ( 2 x + 1 ) ( 12 x - 1 ) = 0 ⇒ 2 x + 1 = 0 or 12 x - 1 = 0 ⇒ x = - 1 2 or x = 1 12 ∴ x = 1 12 , - 1 2 ∵ x = - 1 2 , will not satisfy the given equation Hence, x = 1 12 .

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Important Questions on Inverse Trigonometric Functions

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Prove that:

3 \sin^{- 1} x = \sin^{- 1} (3 x - 4 x^{3}), x \in [- \frac{1}{2}, \frac{1}{2}]

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

\cos^{- 1} x - \cos^{- 1} \frac{y}{2} = α,

where

- 1 \leq x \leq 1, - 2 \leq y \leq 2, x \leq \frac{y}{2},

then for all

x, y, 4 x^{2} - 4 x y \cos α + y^{2}

is equal to :

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Find the value of

\tan^{- 1} (\sqrt{3}) - \cot^{- 1} (- \sqrt{3})

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Prove that

\cos^{- 1} x = 2 \cos^{- 1} (\sqrt{\frac{1 + x}{2}})

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

\sin^{- 1} \frac{3}{5} - \sin^{- 1} \frac{8}{17} =

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Prove that:

\sin^{- 1} (2 x \sqrt{1 - x^{2}}) = 2 \cos^{- 1} x, \frac{1}{\sqrt{2}} \leq x \leq 1

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

$\tan^{- 1} (\frac{x}{y}) - \tan^{- 1} \frac{x - y}{x + y}$ is equal to.

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

An infinite line of charge with uniform line charge density of

1 C m^{- 1}

is placed along the

y

-axis. A charge of

1 C

is placed on the

x

-axis at a distance of

d = 3 m

from the origin. At what distance

(r)

from the origin on the

x

-axis, the total electric field is zero.

N

(Assume

0 < r < d

)

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

A value of

x

satisfying the equation

\sin [\cot^{- 1} (1 + x)] = \cos [\tan^{- 1} x],

is:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

α = {c o s}^{- 1} (\frac{3}{5})

β = {t a n}^{- 1} (\frac{1}{3})

, where

0 < α, β < \frac{π}{2},

then

α - β

is equal to

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Show that

\sin^{- 1} \frac{3}{5} - \sin^{- 1} \frac{8}{17} = \cos^{- 1} \frac{84}{85}

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

x = {s i n}^{- 1} (\sin 10)

and

y = {c o s}^{- 1} (\cos 10),

then

y - x

is equal to:

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Prove that:

\sin^{- 1} \frac{3}{5} - \sin^{- 1} \frac{8}{17} = c o s^{- 1} \frac{84}{75}

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Solve:

\tan^{- 1} 4 x + \tan^{- 1} 6 x = \frac{π}{4}

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The value of

\sin^{- 1} (\frac{12}{13}) - \sin^{- 1} (\frac{3}{5})

is equal to:

EASY

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

\sin^{- 1} (\frac{x}{5}) + {c o s e c}^{- 1} (\frac{5}{4}) = \frac{π}{2},

then the value of

x

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

How is phenol prepared from Aniline? Write the equation.

MEDIUM

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

Which of the following statement is not true about solar cell?

HARD

Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

The following figure shows two identical triode tubes connected in parallel. The anodes are connected together, the grids are connected together and the the cathodes are connected together. Show that the equivalent plate resistance is half of the individual plate resistance, the equivalent mutual conductance is double the individual mutual conductance and the equivalent amplification factors is the same as the individual amplification factor.

Question Image

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Mathematics>Trigonometry>Inverse Trigonometric Functions>Properties of Inverse Trigonometric Functions

How solar energy is used in a spacecraft?

Which of the following travelers provided an account of the Vijayanagara Empire?

Important Questions on Inverse Trigonometric Functions

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