Sketch a right triangle corresponding to the trigonometric function of the acute angle θ. Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of θ.
cot(θ) = 3
sin(θ)=
cos(θ)=
tan(θ)=
csc(θ)=
sec(θ)=
recall that
tanθ = 1/cotθ
sec^2θ = 1 + tan^2θ
csc^2θ = 1 + cot^2θ
sinθ = 1/cscθ
cosθ = 1/secθ
Or, just draw a right triangle with ABC with B=90°
Then AB=3, BC=1
now figure the hypotenuse, and just read off the trig functions from the triangle.
Over to you Oobleck : )
thank you guys
To sketch a right triangle corresponding to the trigonometric function of the acute angle θ and to determine the third side using the Pythagorean Theorem, we need to have the value of one of the sides of the triangle. In this case, the given trigonometric function is cot(θ) = 3.
Cotangent (cot) is the ratio of the adjacent side to the opposite side in a right triangle. Therefore, in the right triangle, the adjacent side will be the known side (which is 3) and the opposite side will be the unknown side.
To find the unknown side, we can use the Pythagorean Theorem, which states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the unknown side, and the two other sides are the adjacent and opposite sides.
Let's assume the adjacent side is 'a', the opposite side is 'b', and the hypotenuse is 'c'. According to the Pythagorean Theorem, we have:
c^2 = a^2 + b^2
Plugging the values we know into the equation, we get:
c^2 = 3^2 + b^2
c^2 = 9 + b^2
Now, let's solve for 'c'. Since 'c' is the hypotenuse, it should be positive, so we can remove the square:
c = sqrt(9 + b^2)
Now that we have the length of the hypotenuse, we can find the other five trigonometric functions of θ.
To do that, we can use the standard trigonometric ratios:
1. sin(θ) = opposite/hypotenuse
In our case, sin(θ) = b/c
2. cos(θ) = adjacent/hypotenuse
In our case, cos(θ) = a/c = 3/c
3. tan(θ) = opposite/adjacent
In our case, tan(θ) = b/a = b/3
4. csc(θ) = 1/sin(θ)
In our case, csc(θ) = 1/(b/c) = c/b
5. sec(θ) = 1/cos(θ)
In our case, sec(θ) = 1/(3/c) = c/3
Now, to find the value of sin(θ), cos(θ), tan(θ), csc(θ), and sec(θ), plug in the value of 'b' we found earlier:
sin(θ) = b/c = b/sqrt(9 + b^2)
cos(θ) = 3/c = 3/sqrt(9 + b^2)
tan(θ) = b/a = b/3
csc(θ) = c/b = sqrt(9 + b^2)/b
sec(θ) = c/3 = sqrt(9 + b^2)/3
Note: The specific values of sin(θ), cos(θ), tan(θ), csc(θ), and sec(θ) depend on the value of 'b', which we cannot determine without additional information.