Question

if cos alpha=8/17 and sin beta=5/13 then find the value of sin (alpha -beta)​

Answer 1

EXPLANATION.

cosα = 8/17.  and  sinβ = 5/13.

By using the Pythagoras theorem,

⇒ H² = P² + B².

(A) = cosα = 8/17 = base/hypotenuse.

⇒ (17)² = P² + (8)².

⇒ 289 = P² + 64.

⇒ 289 - 64 = P².

⇒ 225 = P².

⇒ P = √225.

⇒ P = 15.

sinα = perpendicular/hypotenuse. = 15/17.

cosα = base/hypotenuse. = 8/17.

tanα = perpendicular/base. = 15/8.

cosecα = hypotenuse/perpendicular. = 17/15.

secα = hypotenuse/base. = 17/8.

cotα = base/perpendicular. = 8/15.

(B) = sinβ = perpendicular/hypotenuse. = 5/13.

By using Pythagorean  Theorem,

⇒ H² = P² + B².

⇒ (13)² = (5)² + B².

⇒ 169 = 25 + B².

⇒ 169 - 25 =B².

⇒ 144 = B².

⇒ B = √144.

⇒ B = 12.

sinβ = perpendicular/hypotenuse. = 5/13.

cosβ = base/hypotenuse. = 12/13.

tanβ = perpendicular/base. = 5/12.

cosecβ = hypotenuse/perpendicular = 13/5.

secβ = hypotenuse/base. = 13/12.

cotβ = base/perpendicular. = 12/5.

To find value of Sin ( α - β ).

sin (α - β ) = sinα. cosβ - cosα. sinβ.

sin ( α - β ) = 15/17 X 12/13 - 8/17 X 5/13.

sin ( α - β ) = 180/221 - 40/221.

sin ( α - β )  = 140/221.

                                         

MORE INFORMATION.

(1) = sin2∅ = 2sin∅cos∅ = 2tan∅/1 + tan²∅.

(2) = cos2∅ = cos²∅ - sin²∅ = 2cos²∅ - 1 = 1 - 2sin²∅ = 1 - tan²∅/1 + tan²∅.

(3) = tan2∅ = 2tan∅/ 1 - tan²∅.

Answer 2

Answer:

Solution:-

• First we should make a right angle triangle with cos alpha. And we have to find the value of opposite angle with the help of Pythagoras theorem. After we can find sin alpha value

• And After we should make a other right angle triangle with sin beta. and finding adjacent value after that we can find cos beta value.

• We can find answer By using the formula Sin(α-β) = Sinα×cosβ-cosα×sinβ

From the 1st attachment :-

⇾ AB² + BC² = AC²

⇾ x² + 8² = 17²

⇾ x² + 64 = 289

⇾x² = 289 - 64

⇾x² = 225

⇾x = √225 = 15

⇾x = 15

So, Sinα = 15/17

From the 2nd attachment:-

⇾AB² + BC² = AC²

⇾5² + x² = 13²

⇾25 + x² = 169

⇾x² = 169 - 25

⇾x² = 144

⇾x = √144 = 12

⇾x = 12

So, Cosβ = 12/13

Let's find sin(α-β) :-

${ \implies{ \sf{sin(α-β) = sin \alpha \times cos \beta - cos \alpha \times sin \beta }}}$

${ \implies{ \sf{ \frac{15}{17} \times \frac{12}{13 } - \frac{8}{17} \times \frac{5}{13} }}}$

${ \implies{ \sf{ \frac{180}{221} - \frac{40}{221} }}}$

${ \implies{ \sf{ \frac{180 - 40}{221} }}}$

${ \implies{ \sf{ \bold{ \frac{140}{221} }}}}$

Therefore,

• sin(α-β) = 140/221