The point (3, -4) is on the terminal side of an angle(theta) . What is cos(theta)
To find the value of cos(theta), we can use the coordinates (3, -4) and Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
In this case, we can consider the point (3, -4) as the coordinates of a right triangle, where the x-coordinate 3 is the adjacent side and the y-coordinate -4 is the opposite side from the angle we are interested in.
Using the Pythagorean theorem, we can find the length of the hypotenuse (c) as follows:
c^2 = a^2 + b^2
c^2 = 3^2 + (-4)^2
c^2 = 9 + 16
c^2 = 25
Taking the square root of both sides gives us:
c = sqrt(25)
c = 5
Now, to find cos(theta), we can use the trigonometric ratio:
cos(theta) = adjacent / hypotenuse
cos(theta) = 3 / 5
Therefore, cos(theta) = 3/5.
To find the value of cos(theta), we need to determine the cosine of the angle with respect to the given point (3, -4).
Here's the step-by-step process:
Step 1: Calculate the value of the adjacent side.
The x-coordinate of the given point represents the value of the adjacent side. In this case, the x-coordinate is 3.
Step 2: Calculate the value of the hypotenuse.
To determine the hypotenuse, we can use the distance formula or apply the Pythagorean theorem. However, since only the terminal point is given, we cannot determine the length of the hypotenuse without additional information. Therefore, we will assume that the hypotenuse is the distance between the origin (0, 0) and the given point (3, -4).
Using the distance formula, the hypotenuse can be calculated as follows:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
distance = sqrt((3 - 0)^2 + (-4 - 0)^2)
distance = sqrt(9 + 16)
distance = sqrt(25)
distance = 5
Step 3: Calculate cos(theta).
The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse. Therefore:
cos(theta) = adjacent side / hypotenuse
Now, substitute the values into the formula:
cos(theta) = 3 / 5
So, cos(theta) = 3/5.
draw your triangle. It is in QIV, where x is positive and y is negative.
Now, recall that
r^2 = x^2+y^2 and r is always positive
sinθ = y/r
cosθ = x/r
Now just read off the values from your triangle.