find the exact values of the six trigonometric functions of theta if the terminal side of theta in standard position contains the given point:
1. (8, 4)
2. (4, 4√3)
3. (0, -4)
4. (6, 2)
To find the exact values of the six trigonometric functions of theta when the terminal side of theta in standard position contains a given point, we can use the properties of right triangles and the unit circle.
First, we need to determine the lengths of the sides of the right triangle formed by the given point. Let's go through each case:
1. (8, 4):
To find the lengths of the sides of the right triangle, we can use the Pythagorean theorem. In this case, the hypotenuse is the distance between the point and the origin, which is √(8^2 + 4^2) = √(64 + 16) = √80 = 4√5.
The adjacent side is the x-coordinate of the point, which is 8, and the opposite side is the y-coordinate, which is 4.
Using these values, we can calculate the six trigonometric functions of theta:
- sine (sin) = opposite/hypotenuse = 4/(4√5) = 1/√5
- cosine (cos) = adjacent/hypotenuse = 8/(4√5) = 2/√5
- tangent (tan) = opposite/adjacent = 4/8 = 1/2
- cosecant (csc) = 1/sin = √5
- secant (sec) = 1/cos = √5/2
- cotangent (cot) = 1/tan = 2
2. (4, 4√3):
Similar to the previous case, we find the hypotenuse using the Pythagorean theorem: √(4^2 + (4√3)^2) = √(16 + 48) = √64 = 8.
The adjacent side is 4, and the opposite side is 4√3.
Using these values, we can calculate the trigonometric functions:
- sin = opposite/hypotenuse = (4√3)/8 = √3/2
- cos = adjacent/hypotenuse = 4/8 = 1/2
- tan = opposite/adjacent = (4√3)/4 = √3
- csc = 1/sin = 2/√3 = 2√3/3
- sec = 1/cos = 2
- cot = 1/tan = 1/√3 = √3/3
3. (0, -4):
In this case, the hypotenuse is simply the absolute value of the y-coordinate, which is |-4| = 4.
The adjacent side is 0, and the opposite side is -4.
Using these values, we can calculate the trigonometric functions:
- sin = opposite/hypotenuse = -4/4 = -1
- cos = adjacent/hypotenuse = 0/4 = 0
- tan = opposite/adjacent = -4/0 (undefined)
- csc = 1/sin = -1
- sec = 1/cos = undefined
- cot = 1/tan = 0
4. (6, 2):
Again, using the Pythagorean theorem, we find the hypotenuse: √(6^2 + 2^2) = √(36 + 4) = √40 = 2√10.
The adjacent side is 6, and the opposite side is 2.
Using these values, we can calculate the trigonometric functions:
- sin = opposite/hypotenuse = 2/(2√10) = 1/√10
- cos = adjacent/hypotenuse = 6/(2√10) = 3/√10
- tan = opposite/adjacent = 2/6 = 1/3
- csc = 1/sin = √10
- sec = 1/cos = √10/3
- cot = 1/tan = 3