Question

Angle α lies in quadrant II , and tan α = $-\frac{12}{5}$ . Angle β lies in quadrant IV , and cosβ=3/5 .

What is the exact value of sin(α+β) ?

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sin(α+β) =

Answer 1

Since $\alpha$ lies in quadrant II and $\beta$ lies in quadrant IV, we expect $\sin\alpha>0$, $\cos\alpha<0$, and $\sin\beta<0$.

Recall the Pythagorean identities,

$\sin^2x+\cos^2x=1\iff1+\cot^2x=\csc^2x\iff\tan^2x+1=\sec^2x$

It follows that

$\sec\alpha=\dfrac1{\cos\alpha}=-\sqrt{\tan^2\alpha+1}=-\dfrac{13}5\implies\cos\alpha=-\dfrac5{13}$

$\sin\alpha=\sqrt{1-\cos^2\alpha}=\dfrac{12}{13}$

$\sin\beta=-\sqrt{1-\cos^2\beta}=-\dfrac45$

Recall the angle sum identity for sine:

$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$

So we have

$\sin(\alpha+\beta)=\dfrac{12}{13}\dfrac35+\left(-\dfrac45\right)\left(-\dfrac5{13}\right)=\boxed{\dfrac{56}{65}}$

Answer 2

The value of sin(α+β) is 56/65

Trigonometry identity

Given the following parameters

tan α = -12/5 = opposite/adjacent

Determine the hypotenuse using Pythagoras theorem:

hyp² = 12² + 5²

hyp² = 144 + 25

hyp² = 169

hyp = 13

Determine the value of  sin α and cos α

sin α = opp/hyp

sin α = 12/13

cos α = adj/hyp = -5/13

Similarly if cosβ=3/5 = adj/hyp

opp^2 = 5^2 - 3^2

opp^2 = 16
opp = 4

sin β = opp/hyp = -4/5

Determine the value of sin(α+β)

sin(α+β) = sinαcosβ + cosαsinβ

sin(α+β) = 12/13(3/5) + (-5/13)(-4/5)

sin(α+β) = 56/65

Hence the value of sin(α+β) is 56/65

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