MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Applications of Cross Product

Prove by vect...

Question

Prove by vector method that $sin (α + β) = sin α cos β + cos α sin β$ .

Open in App

Solution

Let $O X$ and $O Y$ be two mutually perpendicular lines intersecting at $O$ .

Let the unit vectors along $O X$ and $O Y$ be $^i$ and $^j$

Let $O B$ be a line above axis $O X$ making angle $α$ with the axis $O X$ .

$∠ A O B = α + β$

Take a point $P$ on $O A$ such that $O P = 1$ , $P M ⊥ O X$

Take a point $Q$ on $O B$ such that $O Q = 1, Q N ⊥ O X$

$\to O Q = \to O N + \to N Q$

$= (O N)^i + (N Q)^j$

$= (O Q cos α)^i + (O Q sin α)^j$

$= (1 cos α)^i + (1 sin α)^j$

$= cos α^i + sin α^j$ .......(1)

$\to O P = \to O M + \to M P$

$= (O M)^i + (- \to N P) (-^j)$

$= (O P cos β)^i + (O P sin β) (-^j)$

$= (1 cos β)^i + (1 sin β) (-^j)$

$= cos β^i - sin β^j$ ........(2)

$\to O P \times \to O Q = (cos β^i - sin β^j) \times (cos α^i + sin α^j)$

$= cos β cos α (^i \times^i) + cos β sin α (^i \times^j) - sin β cos α (^j \times^i) - sin β sin α (^j \times^j)$

$= (sin α cos β + cos α sin β)^k$

Again $\to O P \times \to O Q = 1.1 sin (α + β)^k$

$= sin (α + β)^k$ ........(3)

By equation (2) and (3) we get.
$sin (α + β) = sin α cos β + cos α sin β$ .

Suggest Corrections

thumbs-up

0

Similar questions

Q. Prove by using vectors that

sin (α + β) = sin α cos β + cos α sin β

Q. Using vector, prove that

sin (α - β) = sin α cos β - cos α sin β

Q. Prove by vector method:

sin (α - β) = sin α \cdot cos β - cos α \cdot sin β

.

sin α = \frac{4}{5}

cos β = \frac{12}{13}

sin α cos β + cos α sin β

Q. Prove that: If

0 < α, β, γ < \frac{π}{2}

sin α + sin β + sin γ > sin (α + β + γ)

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Applications of Cross Product

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Applications of Cross Product

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon