The terminal side of an angle in standard position passes through $P(15,-8)$. What is the value of $\sin \theta$?
A. $\sin \theta=-\frac{15}{17}$
B. $\sin \theta=-\frac{8}{17}$
C. $\sin \theta=\frac{8}{17}$
D. $\sin \theta=\frac{15}{17}$
Answer (1)
We are given a point P ( 15 , − 8 ) on the terminal side of an angle θ in standard position.
We calculate the distance r from the origin to the point P using the formula r = x 2 + y 2 , which gives r = 1 5 2 + ( − 8 ) 2 = 17 .
We find sin θ using the formula sin θ = r y , which gives sin θ = 17 − 8 .
The value of sin θ is − 17 8 .
Explanation
Problem Analysis The problem states that the terminal side of an angle θ in standard position passes through the point P ( 15 , − 8 ) . We need to find the value of sin θ .
Finding the distance r To find sin θ , we first need to find the distance r from the origin to the point P ( 15 , − 8 ) . We can use the formula r = x 2 + y 2 , where x = 15 and y = − 8 .
Calculating r Substitute the values of x and y into the formula: r = 1 5 2 + ( − 8 ) 2 r = 225 + 64 r = 289 r = 17
Finding sin(theta) Now that we have the value of r , we can find sin θ using the formula sin θ = r y .
Calculating sin(theta) Substitute the values of y and r into the formula: sin θ = 17 − 8 sin θ = − 17 8
Final Answer Therefore, the value of sin θ is − 17 8 .
Examples
In navigation, if you know the coordinates of a point relative to your starting point, you can use trigonometric functions like sine to determine the direction and angle to that point. For example, if a ship is 15 miles east and 8 miles south of its port, the sine of the angle to the ship helps determine the bearing the ship needs to take to return to port. This is a direct application of trigonometric functions in real-world scenarios.
