Question

if alpha and beta lie in first quadrant and sin alpha= 8/17 , tan beta = 5/12 , find the value of sin ( alpha - beta) , cos ( alpha - beta ) and tan ( alpha - beta k​

Answer 1

Given :-

Alpha ( α ) and Beta ( β ) lie in first quadrant.

◐ sin α = 8/17

◐ tan β = 5/12

To Find :-

• sin ( α - β )
• cos ( α - β )
• tan ( α - β )

Solution :-

Here, we know

→ sin θ = opposite/hypotenuse

→ cos θ = adjacent/hypotenuse

→ tan θ = opposite/adjacent

Let, α and β be the angles of Δ ABC and Δ abc respectively.

Now from given for Δ ABC

→ sin α = 8/17

Hence, opposite = 8 and hypotenuse = 17

so we need to find adjacent

Here we use Pythagoras theorem

→ (hypotenuse)² = (opposite)² + (adjacent)²

→ (17)² = (8)² + (adjacent)²

→ 289 = 64 + (adjacent)²

→ 289 - 64 = (adjacent)²

→ 225 = (adjacent)² i.e.

→ (adjacent)² = 225

→ adjacent = √225

→ adjacent = 15

Now from given for Δ abc

→ tan β = 5/12

Hence, opposite = 5 and adjacent = 12

so we need to find hypotenuse

Here we use Pythagoras theorem

→ (hypotenuse)² = (opposite)² + (adjacent)²

→ (hypotenuse)² = (5)² + (12)²

→ (hypotenuse)² = 25 + 144

→ (hypotenuse)² = 169

→ hypotenuse = √169

→ hypotenuse = 13

Now, for Δ ABC

→ sin α = 8/17

→ cos α = 15/17

→ tan α = 8/15

Now, for Δ abc

→ sin β = 5/13

→ cos β = 12/13

→ tan β = 5/12

Now,

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

→ sin ( α - β ) = 8/17 × 12/13 - 15/17 × 5/13

→ sin ( α - β ) = 96/221 - 75/221

→ sin ( α - β ) = 21/221 or 0.095

→ cos ( α - β ) = cos α × cos β + sin α × sin β

→ cos ( α - β ) = 15/17 × 12/13 + 8/17 × 5/13

→ cos ( α - β ) = 180/221 + 40/221

→ cos ( α - β ) = 220/221 or 0.995

→ tan ( α - β ) = tan α - tan β / 1 + tan α × tan β

→ tan ( α - β ) = 8/15 - 5/12 / 1 + 8/15 × 5/12

→ tan ( α - β ) = 96/180 - 75/180 / 1 + 8/15 × 5/12

→ tan ( α - β ) =21/180 / 1 + 40/180

→ tan ( α - β ) = 21/180 / 220/180

→ tan ( α - β ) = 21/220 or 0.095

Hence,

• sin ( α - β ) = 21/221
• cos ( α - β ) = 220/221
• tan ( α - β ) = 21/220