If θ is an angle in standard position and its terminal side passes through the point (5,-12), find the exact value of
sin
θ
sinθ in simplest radical form.
r^2 - x^2+y^2 = 5^2+(-12)^2 = 169
r = 13.
sin A = y/r = -12/13
A = -67.4o.
To find the exact value of sin(θ) in simplest radical form, we need to use the given information that the terminal side of the angle passes through the point (5, -12).
Let's start by drawing a right triangle with the given coordinates of the terminal point (5, -12):
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(5, -12)
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In the right triangle, the vertical side corresponds to the y-coordinate (-12) and the horizontal side corresponds to the x-coordinate (5). We can use the Pythagorean theorem to find the length of the hypotenuse.
Using the Pythagorean theorem:
hypotenuse² = opposite side² + adjacent side²
hypotenuse² = (-12)² + 5²
hypotenuse² = 144 + 25
hypotenuse² = 169
Taking the square root of both sides, we find:
hypotenuse = √169
hypotenuse = 13
Now, we can calculate the sine of θ. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse of the right triangle. In this case, the opposite side is the y-coordinate (-12), and the hypotenuse is 13:
sin(θ) = opposite / hypotenuse
sin(θ) = -12 / 13
Therefore, the exact value of sin(θ) in simplest radical form is -12/13.
To find the exact value of sinθ, we can use the Pythagorean theorem.
We are given that the point (5,-12) lies on the terminal side of angle θ. Let's locate this point on the coordinate plane:
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-------------- (5,-12)
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The distance from the origin to this point can be found using the Pythagorean theorem:
r = √(x^2 + y^2)
= √(5^2 + (-12)^2)
= √(25 + 144)
= √(169)
= 13
So, we have r = 13.
Now, sinθ is given by the ratio of the y-coordinate (-12) to the radius (13):
sinθ = y / r
= -12 / 13
Therefore, the exact value of sinθ in simplest radical form is -12/13.