Experience the convenience of getting your questions answered at IDNLearn.com. Discover prompt and accurate answers from our community of experienced professionals.
Sagot :
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]
To learn more on trigonometric formulas, we kindly invite to check this verified question: https://brainly.com/question/6904750
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.