knowledge : point (-3,-4)is on the terminal arm of an angel (alpha) in stander position.verify that (sin alpha)2 + (cos alpha )2 =1
FYI: angle, standard
hypotenuse h = √(3^2+4^2) = √25 = 5
sinθ = y/h = -4/5
cosθ = x/h = -3/5
sin^2θ + cos^2θ = (-3/5)^2 + (-4/5)^2 = 9/25 + 16/25 = 25/25 = 1
To verify that (sin α)^2 + (cos α)^2 = 1, we can use the Pythagorean identity:
(sin α)^2 + (cos α)^2 = 1.
Let's find the values of sin α and cos α using the given point (-3, -4) on the terminal arm of α.
First, we need to find the hypotenuse of the right triangle formed by the point (-3, -4). The hypotenuse can be found using the Pythagorean theorem:
hypotenuse = √((-3)^2 + (-4)^2)
= √(9 + 16)
= √25
= 5.
Now, let's find sin α. Since sin α is the ratio of the opposite side to the hypotenuse, we have:
sin α = opposite / hypotenuse
= -4 / 5
= -0.8.
Next, let's find cos α. Since cos α is the ratio of the adjacent side to the hypotenuse, we have:
cos α = adjacent / hypotenuse
= -3 / 5
= -0.6.
Now, let's substitute these values back into the Pythagorean identity:
(sin α)^2 + (cos α)^2 = (-0.8)^2 + (-0.6)^2
= 0.64 + 0.36
= 1.
Therefore, (sin α)^2 + (cos α)^2 = 1. Hence, the given equation is verified.
To verify the equation sin^2(alpha) + cos^2(alpha) = 1 using the given point (-3,-4) on the terminal arm of angle alpha in standard position, we can use the Pythagorean theorem and basic trigonometric definitions.
Step 1: Draw a right triangle on the coordinate plane with the given point (-3,-4) as one of its vertices. The coordinates (-3,-4) represent the triangle's adjacent and opposite sides, respectively. The hypotenuse of the triangle will represent the radius of the unit circle.
Step 2: Calculate the length of the hypotenuse using the Pythagorean theorem. Recall that the Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using the given coordinates (-3,-4), we can calculate the hypotenuse as follows:
Hypotenuse = √((-3)^2 + (-4)^2)
Hypotenuse = √(9 + 16)
Hypotenuse = √25
Hypotenuse = 5
Step 3: Determine the values of sin(alpha) and cos(alpha) using the triangle's sides.
sin(alpha) = opposite / hypotenuse
sin(alpha) = -4 / 5
cos(alpha) = adjacent / hypotenuse
cos(alpha) = -3 / 5
Step 4: Substitute the values of sin(alpha) and cos(alpha) into the equation sin^2(alpha) + cos^2(alpha) = 1 and verify if the equation holds true.
((-4 / 5)^2) + ((-3 / 5)^2) = 16/25 + 9/25 = 25/25 = 1
Therefore, the equation sin^2(alpha) + cos^2(alpha) = 1 is verified for the given point (-3,-4) on the terminal arm of angle alpha in standard position.