Find the values of the trigonometry function of t from the given information tan(t)=1/5 terminal point of t is in quadrant 3

Question

Then sin(t) cost(t) csc (t) sec(t) cot(t)

Answer 1

To find the values of the trigonometric functions for t, we can use the given information that tan(t) = 1/5 and t is in quadrant 3. In quadrant 3, both the sine and cosine functions are negative.

We can start by finding the values of sine and cosine using the tangent:

tan(t) = 1/5
opposite/adjacent = 1/5
opposite = 1, adjacent = 5

Using the Pythagorean theorem, we can find the hypotenuse:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 1^2 + 5^2
hypotenuse^2 = 26
hypotenuse = √26

Now we can find the values of sine and cosine:
sin(t) = opposite/hypotenuse = 1/√26 = √26/26 = √26/26
cos(t) = adjacent/hypotenuse = 5/√26 = 5√26/26 = 5√26/26

Next, we can find the values of csc(t), sec(t), and cot(t) using the reciprocal identities:

csc(t) = 1/sin(t) = 1/(√26/26) = 26/√26 = 26√26/26 = √26
sec(t) = 1/cos(t) = 1/(5√26/26) = 26/(5√26) = 26√26/(5√26) = 26/5
cot(t) = 1/tan(t) = 1/(1/5) = 5

So, the values of the trigonometric functions for t are:
sin(t) = √26/26
cos(t) = 5√26/26
csc(t) = √26
sec(t) = 26/5
cot(t) = 5