Given that α is in Quadrant 1 and tan(α)=67, give an exact answer for the following:
sin(2α)=
cos(2α)=
tan(2α)=
well that seriously helps
opposite/adjacent = 6/7
then hypotenuse = sqrt (36 + 49) = sqrt 85
so sin α = 6/sqrt 85
and cos α = 7/sqrt85
use those in the equations I gave you
is 67 a typo or is it really that close to the y axis?
anyway find sin α and cos α from the tangent then
given sin α / cos α = k (here k = 67)
so sin α = k cos α
sin 2α = 2 sin α cos α = 2 k cos^2 α
cos 2α = cos^2 α - sin^2 α
tan 2α = sin 2α / cos2α
Its a typo. It's suppose to be 6/7 .
To find the values of sin(2α), cos(2α), and tan(2α) given that α is in Quadrant 1 and tan(α) = 67, we can use the double-angle identities for sine, cosine, and tangent.
1. We know that tan(α) = 67. Since α is in Quadrant 1, tan(α) is positive. Therefore, we can find the value of α using the arctan function: α = arctan(67).
2. Once we have found the value of α, we can then find sin(α) and cos(α) using the Pythagorean identity, sin²(α) + cos²(α) = 1. Since α is in Quadrant 1, both sin(α) and cos(α) are positive.
Using the value of α, we can find sin(α) and cos(α) as follows:
sin(α) = sin(arctan(67))
cos(α) = cos(arctan(67))
3. Once we have sin(α) and cos(α), we can use the double-angle identities:
sin(2α) = 2sin(α)cos(α)
cos(2α) = cos²(α) - sin²(α)
tan(2α) = (2tan(α))/(1 - tan²(α))
Let's calculate these values:
α = arctan(67) ≈ 89.388°
sin(α) = sin(89.388°) ≈ 0.9998476
cos(α) = cos(89.388°) ≈ 0.0174524
Next, we can find the values of sin(2α), cos(2α), and tan(2α):
sin(2α) = 2sin(α)cos(α) ≈ 2 * 0.9998476 * 0.0174524 ≈ 0.0348994
cos(2α) = cos²(α) - sin²(α) ≈ 0.0174524² - 0.9998476² ≈ -0.99939
tan(2α) = (2tan(α))/(1 - tan²(α)) ≈ (2 * 67)/(1 - 67²) ≈ 0.0327843
Therefore, the exact values are:
sin(2α) ≈ 0.0348994
cos(2α) ≈ -0.99939
tan(2α) ≈ 0.0327843