The terminal side of theta is in Quadrant III and lies on the line y = 2x. What are the 6 trig functions?
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To find the six trigonometric functions of an angle, we need to know the values of the coordinates (x, y) on the terminal side of the angle. In this case, we are given that the terminal side of theta lies on the line y = 2x in Quadrant III.
In Quadrant III, both x and y coordinates are negative. Using the equation of the line y = 2x, we can substitute -x for x to get y = 2(-x), which simplifies to y = -2x. This means that our coordinates are (-x, -2x), where x is a positive value.
Now, we can use the Pythagorean theorem to determine the magnitude of the vector (the distance from the origin to the point on the line). The Pythagorean theorem states that for any right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse is the distance from the origin to the point (-x, -2x), which can be found using the distance formula or by using the Pythagorean theorem directly. However, since we already know the equation of the line, y = -2x, we can substitute this into the Pythagorean theorem formula to get:
(-x)^2 + (-2x)^2 = hypotenuse^2
x^2 + 4x^2 = hypotenuse^2
5x^2 = hypotenuse^2
Taking the square root of both sides gives us:
sqrt(5x^2) = hypotenuse
sqrt(5)x = hypotenuse
Now that we know the magnitude of the vector (the hypotenuse), we can use it to calculate the trigonometric functions of theta.
1. Sine (sin): The sine function is the ratio of the opposite side to the hypotenuse.
sin(theta) = (opposite side) / (hypotenuse)
= y / hypotenuse
= (-2x) / (sqrt(5)x)
= -2 / sqrt(5)
2. Cosine (cos): The cosine function is the ratio of the adjacent side to the hypotenuse.
cos(theta) = (adjacent side) / (hypotenuse)
= x / hypotenuse
= x / (sqrt(5)x)
= 1 / sqrt(5)
3. Tangent (tan): The tangent function is the ratio of the opposite side to the adjacent side.
tan(theta) = (opposite side) / (adjacent side)
= y / x
= (-2x) / x
= -2
4. Cosecant (csc): The cosecant function is the reciprocal of the sine function.
csc(theta) = 1 / sin(theta)
= 1 / (-2 / sqrt(5))
= -sqrt(5) / 2
5. Secant (sec): The secant function is the reciprocal of the cosine function.
sec(theta) = 1 / cos(theta)
= 1 / (1 / sqrt(5))
= sqrt(5)
6. Cotangent (cot): The cotangent function is the reciprocal of the tangent function.
cot(theta) = 1 / tan(theta)
= 1 / (-2)
= -1/2
Therefore, the trigonometric functions of an angle theta whose terminal side lies on the line y = 2x in Quadrant III are:
sin(theta) = -2 / sqrt(5)
cos(theta) = 1 / sqrt(5)
tan(theta) = -2
csc(theta) = -sqrt(5) / 2
sec(theta) = sqrt(4)
cot(theta) = -1/2