1.If tan theta=3/4 and theta is an acute angle, find cos theta.
2.If sin theta=12/13 and theta is an acute angle, find cot theta.
Both are done the same way, I will do the first, you do the second
tanØ = 3/4 and Ø is acute, thus in the first quadrant
sketch a right-angled triangle, with Ø at the origin , opposite side = 3 and adjacent side = 4
let the hypotenuse be r
r^2 = 4^2 + 3^2 = 25
r = √25 = 5
now you can find any of the 5 remaining trig ratios.
You want cosØ
cosØ = adjacent/hypotenuse = 4/5
1. To find the value of cos theta given that tan theta is 3/4, we can use the identity:
tan^2 theta + 1 = sec^2 theta
In this case, we know that tan theta is 3/4. So, we can substitute this value into the identity:
(3/4)^2 + 1 = sec^2 theta
9/16 + 1 = sec^2 theta
25/16 = sec^2 theta
Now, we can take the square root of both sides to find the value of sec theta:
sqrt(25/16) = sqrt(sec^2 theta)
5/4 = sec theta
Finally, we can use the reciprocal identity to find the value of cos theta:
cos theta = 1 / sec theta
cos theta = 1 / (5/4)
cos theta = 4/5
Therefore, the value of cos theta is 4/5.
2. To find the value of cot theta given that sin theta is 12/13, we can use the identity:
cot theta = 1 / tan theta
In this case, we know that sin theta is 12/13. We can use sin theta and the Pythagorean identity to find the value of cos theta:
sin^2 theta + cos^2 theta = 1
(12/13)^2 + cos^2 theta = 1
144/169 + cos^2 theta = 1
cos^2 theta = 25/169
Now, we can take the square root of both sides to find the value of cos theta:
sqrt(cos^2 theta) = sqrt(25/169)
|cos theta| = 5/13
Since we know the angle theta is acute, we can conclude that cos theta is positive:
cos theta = 5/13
Finally, we can use the reciprocal identity to find the value of cot theta:
cot theta = 1 / tan theta
cot theta = 1 / (12/13)
cot theta = 13/12
Therefore, the value of cot theta is 13/12.