Question

Determine tan(t) if cos(t) = -12/13 and sin(t) ​

Answer 1

Question

Determine tan(t) if cos(t) = -12/13 and sin(t) = 5/13

Answer:

tan(t) =
`(-5)/(12)`

Explanation:

Method 1:

The sketch of the trigonometry is attached to this response.

Since cos(t) is negative and sin(t) is positive, then t should be in the second quadrant as shown in the sketch. A triangle is formed.

We know that, in a given triangle, the cosine of a given angle is the ratio of the adjacent side to the hypotenuse side.

i.e

cos (t) = adjacent / hypotenuse = - 12 / 13

Where;

adjacent = -12 [the adjacent side is in the -x direction]

hypotenuse = 13

Also, the sine of a given angle is the ratio of the opposite side to the hypotenuse side.

sin (t) = opposite / hypotenuse

Where;

opposite = 5

hypotenuse = 13

These sides - opposite, adjacent and hypotenuse - are labelled in the sketch.

From this triangle and the labels, the tangent can be found.

We know that the tangent of a given angle is the ratio of the opposite side to the hypotenuse side. i.e

tan (t) = opposite / adjacent

Where;

opposite = 5

adjacent = -12

Therefore,

tan (t) =
`(-5)/(12)`

Method 2:

The tangent of an angle is the ratio of the sine of that angle to the cosine of that angle. i.e

tan(t) = sin(t) / cos (t) ---------------(i)

sin(t) = 5 / 13

cos(t) = -12 / 13

Substitute for sin(t) and cos (t) in equation (i)

tan(t) = [5 / 13] / [-12 / 13]

tan(t) = [5 / 13] x [-13 / 12]

tan(t) =
`(5)/(13) * (-13)/(12)`

tan(t) =
`(-5)/(12)`