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

Let (-5,4) be a point on the terminal side of theta-example-1
User VietHTran
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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).

User Bhattedon
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