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)
I will do the 2nd one:
for the point (4,4√3)
x = 4, y = 4√3 , r = 8 (by Pythagoras)
sinØ = y/r = 4√3/8
cscØ= 8/(4√3)
cosØ = x/r = 4/8 = 1/2
secØ = 2
tanØ = y/x = 4√3/4 = √3
cotØ = 1/√3
do the others the same way.
To determine the values of the six trigonometric functions (sin, cos, tan, csc, sec, and cot) of a given angle, we need to find the lengths of the sides of a right triangle formed by the given point and the origin. Let's go through each case step by step:
1. (8, 4)
- Step 1: Find the hypotenuse of the right triangle.
- hypotenuse = √(8^2 + 4^2) = √(64 + 16) = √80 = 4√5
- Step 2: Find the values of the trigonometric functions.
- sin(theta) = opposite/hypotenuse = 4/(4√5) = 1/√5
- cos(theta) = adjacent/hypotenuse = 8/(4√5) = 2/√5
- tan(theta) = opposite/adjacent = 4/8 = 1/2
- csc(theta) = 1/sin(theta) = √5
- sec(theta) = 1/cos(theta) = √5/2
- cot(theta) = 1/tan(theta) = 2
2. (4, 4√3)
- Step 1: Find the hypotenuse of the right triangle.
- hypotenuse = √(4^2 + (4√3)^2) = √(16 + 48) = √64 = 8
- Step 2: Find the values of the trigonometric functions.
- sin(theta) = opposite/hypotenuse = (4√3)/8 = √3/2
- cos(theta) = adjacent/hypotenuse = 4/8 = 1/2
- tan(theta) = opposite/adjacent = (√3/2)/(1/2) = √3
- csc(theta) = 1/sin(theta) = 2/√3
- sec(theta) = 1/cos(theta) = 2
- cot(theta) = 1/tan(theta) = 1/√3
3. (0, -4)
- Step 1: Find the hypotenuse of the right triangle.
- hypotenuse = √(0^2 + (-4)^2) = √(0 + 16) = √16 = 4
- Step 2: Find the values of the trigonometric functions.
- sin(theta) = opposite/hypotenuse = -4/4 = -1
- cos(theta) = adjacent/hypotenuse = 0/4 = 0 (undefined)
- tan(theta) = opposite/adjacent = -4/0 (undefined)
- csc(theta) = 1/sin(theta) = -1
- sec(theta) = 1/cos(theta) = undefined
- cot(theta) = 1/tan(theta) = undefined
4. (6, 2)
- Step 1: Find the hypotenuse of the right triangle.
- hypotenuse = √(6^2 + 2^2) = √(36 + 4) = √40 = 2√10
- Step 2: Find the values of the trigonometric functions.
- sin(theta) = opposite/hypotenuse = 2/(2√10) = 1/√10
- cos(theta) = adjacent/hypotenuse = 6/(2√10) = 3/√10
- tan(theta) = opposite/adjacent = 2/6 = 1/3
- csc(theta) = 1/sin(theta) = √10
- sec(theta) = 1/cos(theta) = √10/3
- cot(theta) = 1/tan(theta) = 3
These are the exact values of the six trigonometric functions for each given point.
To find the exact values of the six trigonometric functions of a given point on the terminal side of an angle theta, we can use the properties of trigonometric functions and the Pythagorean theorem.
Let's begin with each given point:
1. (8, 4):
To find the trigonometric functions, we need to determine the values of the opposite, adjacent, and hypotenuse sides.
The hypotenuse (r) can be found using the Pythagorean theorem: r = sqrt(8^2 + 4^2) = 2*sqrt(10).
The opposite side (y) is 4, and the adjacent side (x) is 8.
Now we can find the trigonometric functions:
- sine (sin(theta)) = opposite / hypotenuse = 4 / (2*sqrt(10)) = (2*sqrt(10)) / 10 = sqrt(10) / 5
- cosine (cos(theta)) = adjacent / hypotenuse = 8 / (2*sqrt(10)) = (4*sqrt(10)) / 10 = 2*sqrt(10) / 5
- tangent (tan(theta)) = opposite / adjacent = 4 / 8 = 1 / 2
- cosecant (csc(theta)) = 1 / sin(theta) = 5 / sqrt(10)
- secant (sec(theta)) = 1 / cos(theta) = 5 / (2*sqrt(10))
- cotangent (cot(theta)) = 1 / tan(theta) = 2
2. (4, 4√3):
Similar to the previous example, we need to determine the values of the opposite, adjacent, and hypotenuse sides.
The hypotenuse (r) can be found using the Pythagorean theorem: r = sqrt(4^2 + (4√3)^2) = sqrt(16 + 48) = sqrt(64) = 8.
The opposite side (y) is 4√3, and the adjacent side (x) is 4.
Now we can find the trigonometric functions:
- sin(theta) = 4√3 / 8 = √3 / 2
- cos(theta) = 4 / 8 = 1 / 2
- tan(theta) = (4√3) / 4 = √3
- csc(theta) = 1 / sin(theta) = 2 / √3 = 2√3 / 3
- sec(theta) = 1 / cos(theta) = 2
- cot(theta) = 1 / tan(theta) = 1 / √3 = √3 / 3
3. (0, -4):
In this case, the angle theta lies on the negative y-axis, so the adjacent side (x) is 0. The opposite side (y) is -4.
The hypotenuse (r) can be found using the Pythagorean theorem: r = sqrt(0^2 + (-4)^2) = 4.
Now we can find the trigonometric functions:
- sin(theta) = -4 / 4 = -1
- cos(theta) = 0 / 4 = 0
- tan(theta) = -4 / 0 (undefined)
- csc(theta) = 1 / sin(theta) = -1
- sec(theta) = 1 / cos(theta) (undefined)
- cot(theta) = 1 / tan(theta) (undefined)
4. (6, 2):
Similar to the previous examples, we need to determine the values of the opposite, adjacent, and hypotenuse sides.
The hypotenuse (r) can be found using the Pythagorean theorem: r = sqrt(6^2 + 2^2) = sqrt(36 + 4) = sqrt(40) = 2√10.
The opposite side (y) is 2, and the adjacent side (x) is 6.
Now we can find the trigonometric functions:
- sin(theta) = 2 / (2√10) = 1 / √10
- cos(theta) = 6 / (2√10) = 3 / √10 = 3√10 / 10
- tan(theta) = 2 / 6 = 1 / 3
- csc(theta) = 1 / sin(theta) = √10
- sec(theta) = 1 / cos(theta) = 10 / (3√10) = 10 / (3√10) * (√10 / √10) = 10√10 / 30 = √10 / 3
- cot(theta) = 1 / tan(theta) = 3
Remember to simplify the expressions if possible by rationalizing the denominators and reducing fractions when necessary.