Determine the values of sin θ, cos θ, tan θ, csc θ, sec θ, and cot θ at (-3, -4) on the terminal arm of an angle θ in standard position. Special Triangles

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embizze | High School Teacher | (Level 2) Educator Emeritus

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On a standard Cartesian coordinate system draw a triangle whose vertices are at the origin A(0,0), C(-3,0) and B(-3,-4). You will notice that the angle at vertex C is a right angle.

Let `theta` be `/_CAB` . The leg adjacent to `theta` is `bar(CA)` while the leg opposite `theta` is `bar(CB)` . The hypotenuse of the right triangle is `bar(AB)` .

The trigonometric ratios for an acute angle of a right triangle are :

`sin theta=("opp")/("hyp"),cos theta=("adj")/("hyp"),tan theta = "opp"/("adj")` . Also, `sec` is the reciprocal of `cos` , `csc` is the reciprocal of `sin` , while `cot` is the reciprocal of `tan` .

We see that `AC=3,BC=4` and from the Pythagorean theorem `AB=5` .

For the trigonometric ratios, we keep the sign -- that is the side adjacent to `theta` is -3 -- this allows us to identify the quadrant that the angle is located in.


`sin theta="opp"/"hyp" =-4/5`

`cos theta="adj"/"hyp"=-3/5`

`tan theta="opp"/"adj"=4/3`

`cot theta="adj"/"opp"=3/4`

`sec theta="hyp"/"adj"=-5/3`

`csc theta="hyp"/"opp"=-5/4`