IF delta SIN=12/13
find the other five trigonometric functions value of delta
If delta SIN=12/13 mean:
sin Δ = 12 / 13
Then:
cos Δ = ± √ ( 1 - sin² Δ )
cos Δ = ± √ [ 1 - ( 12 / 13 )² ]
cos Δ = ± √ [ 1 - ( 144 / 169 ) ]
cos Δ = ± √ ( 169 / 169 - 144 / 169 )
cos Δ = ± √ ( 25 / 169 )
cos Δ = ± 5 / 13
tan Δ = ± sin Δ / cos Δ
tan Δ = ± ( 12 / 13 ) / ( 5 / 13 ) = ± ( 12 * 13 ) / ( 5 * 13 ) = ± 12 / 5
cot Δ = ± cos Δ / sin Δ = ± ( 5 / 13 ) / ( 12 / 13 ) = ± ( 5 * 13 ) / ( 12 * 13 ) = ± 5 / 12
sec Δ = ± 1 / cos Δ = ± 13 / 5
csc Δ = ± 1 / sin Δ = ± 13 / 12
you have to consider the CAST rule
using Bosnians answers, we have to eliminate some of the ± cases
since sin Δ = 12 / 13
Δ could be in quadrants I or II
in quad I
sinΔ= 12/13 , cscΔ = 13/12
cosΔ = 5/13 , secΔ = 13/5
tanΔ = 12/5 , cotΔ = 5/12
in quad II
sinΔ = 12/13 , cscΔ = 13/12
cosΔ = -5/13 , secΔ = -13/5
tanΔ = -12/5 , cotΔ = -5/12
To find the other five trigonometric functions (cosine, tangent, cosecant, secant, and cotangent) given the value of sin(delta), you can use the following steps:
Step 1: Identify the given value
Given: sin(delta) = 12/13
Step 2: Determine the values of the adjacent and hypotenuse sides of the right triangle
The value of sin(delta) is equal to the ratio of the length of the opposite side to the hypotenuse in a right triangle. Since we know sin(delta) = 12/13, we can let the opposite side be 12 and the hypotenuse be 13.
Step 3: Use the Pythagorean theorem to find the length of the adjacent side
In a right triangle, the Pythagorean theorem states that the sum of the squares of the lengths of the two legs (sides adjacent to the right angle) is equal to the square of the length of the hypotenuse. So, in this case, we can calculate the length of the adjacent side (let's call it a) using the formula:
a = √(hypotenuse^2 - opposite^2)
a = √(13^2 - 12^2)
a = √(169 - 144)
a = √25
a = 5
Now, we have the lengths of the opposite side (12), adjacent side (5), and hypotenuse (13) of the right triangle.
Step 4: Calculate the other trigonometric function values using the given sides
cos(delta) = adjacent / hypotenuse
cos(delta) = 5 / 13
tan(delta) = opposite / adjacent
tan(delta) = 12 / 5
cosec(delta) = 1 / sin(delta)
cosec(delta) = 1 / (12 / 13)
cosec(delta) = 13 / 12
sec(delta) = 1 / cos(delta)
sec(delta) = 1 / (5 / 13)
sec(delta) = 13 / 5
cot(delta) = 1 / tan(delta)
cot(delta) = 1 / (12 / 5)
cot(delta) = 5 / 12
Therefore, the values of the other five trigonometric functions of delta are:
cos(delta) = 5/13
tan(delta) = 12/5
cosec(delta) = 13/12
sec(delta) = 13/5
cot(delta) = 5/12