The correct option is [tex]\boxed{{\mathbf{Option B}}}[/tex] .
Further explanation:
The sine ratio is the ratio of the length of the side that is opposite to the given angle to the length of the hypotenuse.
It can be expressed as,
[tex]\sin \theta =\frac{p}{h}[/tex]
Pythagoras theorem is always used in a right angle tri
angle to obtain the one of the side of the triangle.
Hypotenuse is the longest side in the right angle triangle.
Pythagoras theorem can be expressed as,
[tex]{H^2}={P^2}+{B^2}[/tex]
Here, [tex]H[/tex] is the hypotenuse, [tex]B[/tex] is the base and [tex]P[/tex] is the perpendicular.
Step by step explanation:
Step 1:
The terminal side of an angle passes through the point [tex]p\left({15,-8}\right)[/tex] as shown in the attached figure.
It can be observed that the given point lies in the fourth quadrant as [tex]x[/tex] coordinate is positive and [tex]y[/tex] coordinate is negative.
As we can see in the attached figure the perpendicular is the side of the triangle that is opposite the angle [tex]\theta[/tex] .
Therefore, the perpendicular is 8 and the base is [tex]15[/tex] .
Now find the hypotenuse by the use of Pythagoras theorem.
[tex]\begin{gathered}{H^2}={8^2}+{15^2}\hfill\\{H^2}=64+225\hfill\\H=\sqrt{289}\hfill\\H=17\hfill\\\end{gathered}[/tex]
Therefore, the length of the hypotenuse is [tex]{\text{17 units}}[/tex] .
We know that the sine function is negative in fourth quadrant.
Therefore, [tex]\sin \theta[/tex] cab be expressed as,
[tex]\sin \theta=-\frac{8}{{17}}[/tex]
Thus, option B [tex]\sin \theta=-\frac{8}{{17}}[/tex] is correct.
Learn more:
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Triangles
Keywords: side, lengths, distance, Pythagoras theorem, triangle, hypotenuse, base, perpendicular, right angle triangle, longest side, sine function, fourth quadrant, positive sign, negative sign, ratio.
