Find `cos theta,tan theta` given that `sin theta=-5/7` :
Since the sine is negative, the angle is either in the third quadrant or the fourth quadrant.
Also, in a right triangle the sine is defined as the ratio of the leg opposite the angle divided by the hypotenuse.
(1) If the angle is in the third quadrant, we have the vertical leg as 5 units and the hypotenuse 7 units. Using the pythagorean theorem we find the horizontal leg: `a^2+(-5)^2=7^2==>a^2=24 ==> a=2sqrt(6)` . As this is the third quadrant the leg is `-2sqrt(6).`
` ` Then we use cosine of an angle is the ratio of the leg adjacent to the hypotenuse: `cos theta = (-2sqrt(6))/7`
The tangent is the ratio of the leg opposite the angle to the leg adjacent to the angle so `tan theta=5/(2sqrt(6))=(5sqrt(6))/12`
(2) If the angle is in the fourth quadrant, the numerical values will remain the same, but the cosine will be positive and the tangent will be negative.