The value of the sine of the sum of angles [tex]\theta[/tex] and [tex]\phi[/tex] is [tex]\frac{44}{125}[/tex].
Procedure - Determine of the sine of a sum of angles based on two trigonometric expressions and quadrants.
By trigonometry we know that the sine of a sum of angles is defined by the following formula:
[tex]\sin (\theta + \phi) = \sin \theta \cdot \cos \phi + \cos \theta \cdot \sin \phi[/tex] (1)
In addition we know that sine is positive in the second quadrant and cosine is negative in the second and third quadrants. Besides, tangent is negative in the second quadrant.
By definitions of sine, cosine and tangent we have the following expressions:
[tex]\sin \alpha = \frac{y}{\sqrt{x^{2}+y^{2}}}[/tex] (2)
[tex]\cos \alpha = \frac{x}{\sqrt{x^{2}+y^{2}}}[/tex] (3)
[tex]\tan \alpha = \frac{y}{x}[/tex] (4)
Determination of the sine of the sum of angles
(θ: [tex]x = -4, y = -3[/tex], φ: [tex]x: -24[/tex], [tex]y = 7[/tex])
By (2) we have the following result:
[tex]\sin (\theta + \phi) = \left(-\frac{3}{5} \right)\cdot \left(-\frac{24}{25} \right)+\left(-\frac{4}{5} \right)\cdot \left(\frac{7}{25} \right)[/tex]
[tex]\sin (\theta + \phi) = \frac{44}{125}[/tex]
The value of the sine of the sum of angles [tex]\theta[/tex] and [tex]\phi[/tex] is [tex]\frac{44}{125}[/tex]. [tex]\blacksquare[/tex]
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