Theta is acute and alpha is obtuse . It is known that sin theta=2/5 and cosec alpha =2. Compute the exact values of
a) sec theta
b) cos alpha
c) tan( theta + alpha)
I will use θ for theta, and a for alpha
sin θ = 2/5 , so θ must be in quadrant I
cosec a =2 , so sin a = 1/2, but a is to be obtuse, so a is in II
if sinθ = 2/5 , using a right-angled triangle, cosθ = √21/5, and secθ = 5/√21
if sin a = 1/2, then cos a = √3/2
tan(θ+a) = sin(θ+a) / cos(θ+a)
use the expansion for sin(A+B) and cos(A+B) to find the numerator and denominator
of the right side, then evaluate.
Thanks much
To solve this problem, we will need to use trigonometric identities and the given information to find the exact values of sec theta, cos alpha, and tan( theta + alpha). Here's how you can compute each value:
a) sec theta:
To find the value of sec theta, we can use the reciprocal identity, sec theta = 1/cos theta.
Since sin theta = 2/5, we can use the Pythagorean identity, sin^2 theta + cos^2 theta = 1, to find cos theta.
Here's how we can compute it:
sin^2 theta + cos^2 theta = 1
(2/5)^2 + cos^2 theta = 1
4/25 + cos^2 theta = 1
cos^2 theta = 1 - 4/25
cos^2 theta = 21/25
Taking the square root of both sides, we get:
cos theta = ± sqrt(21)/5
Since theta is acute, the value of cos theta must be positive, so we have:
cos theta = sqrt(21)/5
Now we can find sec theta:
sec theta = 1/cos theta
sec theta = 1 / (sqrt(21)/5)
sec theta = 5 / sqrt(21)
To rationalize the denominator, we multiply the numerator and the denominator by sqrt(21):
sec theta = (5 * sqrt(21)) / 21
b) cos alpha:
We are given that cosec alpha = 2. To find cos alpha, we can use the reciprocal identity, cosec alpha = 1/sin alpha.
Since cosec alpha = 2, we can find sin alpha by taking the reciprocal:
sin alpha = 1/cosec alpha
sin alpha = 1/2
Using the Pythagorean identity, sin^2 alpha + cos^2 alpha = 1, we can find cos alpha:
sin^2 alpha + cos^2 alpha = 1
(1/2)^2 + cos^2 alpha = 1
1/4 + cos^2 alpha = 1
cos^2 alpha = 1 - 1/4
cos^2 alpha = 3/4
Taking the square root of both sides, we get:
cos alpha = ± sqrt(3)/2
Since alpha is obtuse, the value of cos alpha must be negative. Therefore, we have:
cos alpha = -sqrt(3)/2
c) tan( theta + alpha):
To find the value of tan( theta + alpha), we can use the sum identity, tan( theta + alpha) = ( tan theta + tan alpha ) / (1 - tan theta * tan alpha).
We are already given the values of sin theta and cos alpha, so we can find tan theta and tan alpha using the respective identities:
tan theta = sin theta / cos theta
tan theta = (2/5) / (sqrt(21)/5)
tan theta = 2 / sqrt(21)
tan alpha = sin alpha / cos alpha
tan alpha = (1/2) / (-sqrt(3)/2)
tan alpha = -1 / sqrt(3)
tan alpha = -sqrt(3) / 3
Now, we can find tan( theta + alpha):
tan( theta + alpha) = ( tan theta + tan alpha ) / (1 - tan theta * tan alpha)
tan( theta + alpha) = ( 2 / sqrt(21) + (-sqrt(3) / 3) ) / (1 - (2 / sqrt(21)) * (-sqrt(3) / 3))
To simplify this expression further, we can rationalize the denominator and combine the terms in the numerator:
tan( theta + alpha) = (6 - sqrt(63))/(3sqrt(21) - 2sqrt(3))
Therefore, the exact value of tan( theta + alpha) is:
tan( theta + alpha) = (6 - sqrt(63))/(3sqrt(21) - 2sqrt(3))