Determine the exact value if cot(3pi/4) using special triangles.
I don't understand how it could be 1/tan3pi/4 in a triangle....
for any x,
cot(x) = 1/tan(x)
3pi/4 is in Quadrant II, so draw the triangle.
x = -1
y = 1
cot(3pi/4) = x/y = -1/1 = -1
or, consider that
cot(pi-x) = -cot(x)
since cos(pi-x) = -cos(x)
and sin(pi-x) = sin(x)
so, cot(3pi/4) = cot(pi - pi/4) = -cot(pi/4)
To determine the exact value of cot(3π/4) using special triangles, we need to work with the relationship between cotangent and tangent.
In a right triangle, the cotangent of an angle is equal to the reciprocal of the tangent of that angle. So, we can express cot(θ) as 1 / tan(θ).
Now, let's consider the special triangle known as the 45-45-90 triangle. In this triangle, both acute angles are 45 degrees, and the hypotenuse is √2 times the length of one of the legs.
With this information in mind, we can find the value of cot(3π/4) by finding the value of tan(3π/4) and then taking its reciprocal.
In a 45-45-90 triangle, the tangent of one of the acute angles is always equal to 1. So, tan(π/4) = 1.
Now, to find tan(3π/4), we can use the periodicity property of tangent. Tangent has a period of π, so tan(3π/4) is equal to tan(3π/4 - 2π), which simplifies to tan(-π/4).
Since the tangent function is odd, tan(-θ) = -tan(θ), and thus, tan(-π/4) = -tan(π/4) = -1.
Taking the reciprocal of -1, we find that cot(3π/4) = -1.
So, using the relationship between cotangent and tangent in a right triangle, and considering the special triangle of a 45-45-90 triangle, we can determine that the exact value of cot(3π/4) is -1.
To determine the exact value of cot(3π/4) using special triangles, let's start by understanding the relationships between the trigonometric functions cosine (cos), sine (sin), tangent (tan), and cotangent (cot).
In a right triangle, cosine is defined as the ratio of the length of the adjacent side to the length of the hypotenuse:
cos(θ) = adjacent / hypotenuse
Sine is defined as the ratio of the length of the opposite side to the length of the hypotenuse:
sin(θ) = opposite / hypotenuse
Tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side:
tan(θ) = opposite / adjacent
And cotangent is defined as the reciprocal of tangent:
cot(θ) = 1 / tan(θ)
Now, let's consider the special triangle known as the 45-45-90 triangle. In this triangle, the two acute angles are both 45 degrees, and the sides have specific length ratios. The hypotenuse is longer than the legs, and the legs are congruent to each other.
In a 45-45-90 triangle, the ratio of the length of the leg to the length of the hypotenuse is √2 / 2. Therefore, we can determine the value of cot(π/4) using this triangle.
cot(π/4) = 1 / tan(π/4)
Since tan(π/4) = sin(π/4) / cos(π/4), and sin(π/4) = √2 / 2 and cos(π/4) = √2 / 2, we can substitute these values into the equation:
cot(π/4) = 1 / (√2 / 2) = 2 / √2 = √2
So, the exact value of cot(π/4) using special triangles is √2.