Assume that alpha is an angle in the standard position whose terminal side contains the given point. Find the exact values of sin alpha, cos alpha, tan alpha, csc alpha, sec alpha, and cot alpha. (3,4)
To find the exact values of the trigonometric functions, we need to determine the lengths of the legs of the right triangle with the given point (3, 4) as its endpoint. The hypotenuse will be the distance from the origin to the given point, which can be found using the Pythagorean theorem.
Using the distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
d = √((3 - 0)² + (4 - 0)²)
d = √(3² + 4²)
d = √(9 + 16)
d = √25
d = 5
Now that we have the length of the hypotenuse, we can find the values of the trigonometric functions:
sin α = opposite/hypotenuse = 4/5
cos α = adjacent/hypotenuse = 3/5
tan α = opposite/adjacent = 4/3
csc α = 1/sin α = 5/4
sec α = 1/cos α = 5/3
cot α = 1/tan α = 3/4
Therefore:
sin α = 4/5
cos α = 3/5
tan α = 4/3
csc α = 5/4
sec α = 5/3
cot α = 3/4