Given tan(α)=√3/3 and 0<α<π/2, find the exact values of the remaining five trigonometric functions.
hypotenuse = sqrt (3+9)= sqrt 12
= 2 sqrt 3
sin = sqrt 3 / hypotenuse
cos = 3/hypotenuse
etc
ty!
To find the exact values of the remaining five trigonometric functions, we can make use of the given tangent value and the fact that tangent is the ratio of sine to cosine.
First, let's recall the definitions of the trigonometric functions:
- Sine (sin): the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos): the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan): the ratio of the length of the opposite side to the length of the adjacent side.
- Cosecant (csc): the reciprocal of the sine.
- Secant (sec): the reciprocal of the cosine.
- Cotangent (cot): the reciprocal of the tangent.
Since we are given the value of tangent (tan(α) = √3/3), we can use the Pythagorean identity to find the values of sine and cosine.
The Pythagorean identity states that for any angle α in a right triangle:
sin^2(α) + cos^2(α) = 1
By substituting the given value of tangent and squaring it, we get:
(sin(α) / cos(α))^2 = 3/3
Now, since we are given that 0 < α < π/2, we know that α is in the first quadrant. In the first quadrant, both sine and cosine are positive.
Let's solve for sine (sin(α)):
sin^2(α) = (3/3) * cos^2(α)
sin^2(α) = cos^2(α)
sin(α) = ± cos(α)
Since sine and cosine are positive in the first quadrant, we can take the positive square root:
sin(α) = cos(α)
Now, we know that sine and cosine have the same value. Using the Pythagorean identity, we can solve for their value:
sin^2(α) + cos^2(α) = 1
sin^2(α) + sin^2(α) = 1
2sin^2(α) = 1
sin^2(α) = 1/2
sin(α) = ±√(1/2)
Again, since sine is positive in the first quadrant, we take the positive square root:
sin(α) = √(1/2) = √2/2
Now that we know the values of sine and cosine, we can find the values of the remaining trigonometric functions using the definitions:
- Cosecant (csc) = 1 / sin(α)
csc(α) = 1 / (√2/2) = 2 / √2 = √2
- Secant (sec) = 1 / cos(α)
sec(α) = 1 / (√2/2) = 2 / √2 = √2
- Cotangent (cot) = 1 / tan(α)
cot(α) = 1 / (√3/3) = 3 / √3 = √3
Therefore, the exact values of the remaining trigonometric functions are:
sin(α) = √2/2
cos(α) = √2/2
csc(α) = √2
sec(α) = √2
cot(α) = √3