Question

Let (-5,4) be a point on the terminal side of theta

Answer 1

If point (-5,4) is a point on the terminal side of
`\theta`, then angle
`\theta` is obtuse (point (-5,4) is placed in the second quadrant). Consider right triangle with legs 5 and 4 and angle
`\theta_1` (
`\theta_1` is in the first quadrant) adjacent to the leg with length 5. By the Pythagorean theorem
`\text{ hypotenuse }^2=4^2+5^2=41` and the length of the
`\text{ hypotenuse }=√(41)`. Then:

1.
`\sin \theta_1=\frac{\text{ opposite leg }}{\text {hypotenuse }} =(4)/(√(41))`;

2.
`\sec \theta_1=(1)/(\cos \theta_1)=\frac{\text{ hypotenuse }}{\text{ adjacent leg }}=(√(41))/(5)`;

3.
`\tan \theta_1=\frac{\text{opposite leg}}{\text{adjacent leg}}=(4)/(5)`.

In the second quadrant function
`\sin` has the same sign as in the first quadrant and functions
`\sec, \tan` have the opposite sign.

Therefore,

1.
`\sin \theta=(4)/(√(41))`;

2.
`\sec \theta =-(√(41))/(5)`;

3.
`\tan \theta =-(4)/(5)`.