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) = 9
cot(θ) = 9
or
tan(θ) = 1/9
recall that tan θ = opposite/adjacent
so draw a right-angled triangle with opposite 1 and adjacent 9
find the hypotenuse h
h^2 = 1^2 + 9^2 = 82
h = √82
sin θ = 1/√82 , cscθ = √82
cosθ = 9/√82 , secθ = √82/9
tanθ = 1/9 , cotθ = 9
thanks reiny
To sketch a right triangle corresponding to the trigonometric function cot(θ) = 9, we can start by labeling the sides of the triangle.
Let's assume that the angle θ is acute, which means it will be one of the acute angles in the right triangle.
We know that cot(θ) = 9, which is the ratio of the adjacent side to the opposite side. Therefore, we can label the adjacent side as 9 and the opposite side as 1 (since cot(θ) = adjacent/opposite).
Using the Pythagorean Theorem (a^2 + b^2 = c^2), we can find the length of the hypotenuse (c). In this case, c is the unknown side.
Using the given values, we have:
9^2 + 1^2 = c^2
81 + 1 = c^2
82 = c^2
c = √82 ≈ 9.06 (rounded to 2 decimal places)
Now that we have determined the lengths of all three sides (the adjacent side = 9, the opposite side = 1, and the hypotenuse ≈ 9.06), we can find the other trigonometric functions of θ.
To find sine, cosine, secant, cosecant, and tangent, we can use the following formulas:
sine(θ) = opposite/hypotenuse
cosine(θ) = adjacent/hypotenuse
tangent(θ) = opposite/adjacent
cosecant(θ) = 1/sine(θ)
secant(θ) = 1/cosine(θ)
Substituting the corresponding values, we get:
sine(θ) = 1/9.06 ≈ 0.11
cosine(θ) = 9/9.06 ≈ 0.99
tangent(θ) = 1/9 ≈ 0.11
cosecant(θ) = 1/(1/9.06) ≈ 9.06
secant(θ) = 1/(9/9.06) ≈ 1.01
Therefore, the other trigonometric functions of θ are:
sine(θ) ≈ 0.11
cosine(θ) ≈ 0.99
tangent(θ) ≈ 0.11
cosecant(θ) ≈ 9.06
secant(θ) ≈ 1.01
To sketch a right triangle corresponding to the trigonometric function, we need to understand what cotangent represents.
The cotangent (cot) function is the ratio of the adjacent side to the opposite side of a right triangle in relation to an acute angle θ.
In this case, we are given that cot(θ) = 9. To find the other trigonometric functions, we'll need to determine the adjacent, opposite, and hypotenuse sides of the right triangle.
Using the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can determine the third side.
Let's assume the adjacent side is a and the opposite side is b. Therefore, we have cot(θ) = a / b = 9.
From this, we can conclude that a = 9b.
Now, let's use the Pythagorean Theorem to find the hypotenuse:
a^2 + b^2 = c^2
Substituting the value of a, we have (9b)^2 + b^2 = c^2
81b^2 + b^2 = c^2
82b^2 = c^2
Taking the square root of both sides, we get:
√(82b^2) = √c^2
√(82) * b = c
Now we have determined all three sides of our right triangle. We have a = 9b, b, and c = √(82) * b.
To find the other trigonometric functions, we can use the following formulas:
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
We are given cot(θ) = 9, which is the reciprocal of tan(θ). Therefore:
tan(θ) = 1/9
Using this value, we can find the other trigonometric functions:
sin(θ) = 1/csc(θ) = 1/√(1 + tan^2(θ)) = 1/√(1 + (1/9)^2)
cos(θ) = 1/sec(θ) = 1/√(1 + cot^2(θ)) = 1/√(1 + 9^2)
tan(θ) = 1/9
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
Now, substitute the values of sin(θ), cos(θ), csc(θ), and sec(θ) into the equations to find their respective values.