If point (3, -8) lies on the terminal side of an angle A in standard position, find:
a) sinA
b) tanA
if 0 is an angle in standard position and its terminal side passes through the point (-3,2) find the exact value of csc 0
To find the values of sinA and tanA, we need to determine the values of the opposite side, adjacent side, and hypotenuse of the right triangle formed by the angle A in standard position.
Given that point (3, -8) lies on the terminal side of angle A, we can draw a line segment from the origin (0,0) to point (3, -8). This line segment will serve as the hypotenuse of the right triangle.
To calculate the lengths of the opposite side and adjacent side, we can use the coordinates of point (3, -8). The y-coordinate (-8) represents the opposite side, while the x-coordinate (3) represents the adjacent side.
a) sinA is equal to the ratio of the opposite side to the hypotenuse:
sinA = opposite/hypotenuse = -8/hypotenuse
To determine the length of the hypotenuse, we can use the Pythagorean theorem:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = (-8)^2 + 3^2
hypotenuse^2 = 64 + 9
hypotenuse^2 = 73
hypotenuse = √73
Plugging this value into the equation for sinA:
sinA = -8/√73
b) tanA is equal to the ratio of the opposite side to the adjacent side:
tanA = opposite/adjacent = -8/3
Therefore, the values are:
a) sinA = -8/√73
b) tanA = -8/3
To find the values of the trigonometric functions for angle A, we can use the coordinates of the point (3, -8).
First, let's represent the point (3, -8) in a coordinate system. The x-coordinate represents the horizontal distance from the origin, and the y-coordinate represents the vertical distance.
In this case, the point (3, -8) is in the fourth quadrant since both coordinates are positive.
a) To find sinA, we need to identify the ratio of the y-coordinate (-8) to the hypotenuse. However, we first need to find the length of the hypotenuse using the Pythagorean theorem.
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Applying the theorem, we have:
hypotenuse^2 = (x-coordinate)^2 + (y-coordinate)^2
= 3^2 + (-8)^2
= 9 + 64
= 73
Taking the square root of both sides, we get:
hypotenuse = sqrt(73)
Now, we can calculate sinA:
sinA = opposite/hypotenuse = y-coordinate/hypotenuse
= -8/sqrt(73)
b) To find tanA, we need to identify the ratio of the y-coordinate to the x-coordinate.
tanA = opposite/adjacent = y-coordinate/x-coordinate
= -8/3
Therefore, for the point (3, -8) on the terminal side of angle A in standard position:
a) sinA = -8/sqrt(73)
b) tanA = -8/3
3^2 + 8^2 = h^2
9 + 64 = h^2
73 = h^2
h = sqrt 73
a) sin A = -8/sqrt(73)
b) tan A = -8/3