Prove that:
$\cos ^{-1} \frac{4}{5}+\sin ^{-1} \frac{3}{5}=\sin ^{-1} \frac{27}{11}$
To Prove: $\cos ^{-1} \frac{4}{5}+\sin ^{-1} \frac{3}{5}=\sin ^{-1} \frac{27}{11}$
Formula Used: $\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \times \sqrt{1-y^{2}}+y \times \sqrt{1-x^{2}}\right)$
Proof:
$\mathrm{LHS}=\cos ^{-1} \frac{4}{5}+\sin ^{-1} \frac{3}{5} \ldots(1)$
Let $\cos \theta=\frac{4}{5}$
Therefore $\theta=\cos ^{-1} \frac{4}{5} \ldots$ (2)
From the figure, $\sin \theta=\frac{3}{5}$
$\Rightarrow \theta=\sin ^{-1} \frac{3}{5} \ldots$ (3)
From (2) and (3),
$\cos ^{-1} \frac{4}{5}=\sin ^{-1} \frac{3}{5}$
Substituting in (1), we get
$\mathrm{LHS}=\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{3}{5}$
$=\sin ^{-1}\left(2 \times \frac{3}{5} \times \sqrt{1-\left(\frac{3}{5}\right)^{2}}\right)$
$=\sin ^{-1}\left(2 \times \frac{3}{5} \times \sqrt{1-\frac{9}{25}}\right)$
$=\sin ^{-1}\left(2 \times \frac{3}{5} \times \sqrt{\frac{16}{25}}\right)$
$=\sin ^{-1}\left(2 \times \frac{3}{5} \times \frac{4}{5}\right)$
$=\sin ^{-1} \frac{24}{25}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,