Find the value of sin ø, if tan ø = 4; 180°<ø<270°

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Answer 1

Aubrijay,

The given condition: tan(phi) = 4 implies a right triangle with a ratio for y/x of 4. Right?

All you have to do is draw a diagram of your situation. A situation in a trig problem means a triangle (or more than one!). So go into the 3rd quadrant of the x-y plane, and draw a triangle, with one vertex at (0,0), one at (-1,0), and one at (-1,-4), with sides y=4, x=1 (actually, each of these is a negative number, right? More on this below, but for now, just make sure the triangle is drawn to scale). Now, what side of the triangle don't you have a value for? The hypotenuse, right? So, find it, using the Pythagorean theorem. Now you can use this same triangle to find the sin of the angle (the angle, incidentally, is the angle opened between the ray from (0,0) to the right along the x-axis, around to the ray from (0,0)  through (-1,-4) ).

How do you find the sin? = y/r, right? Just substitute the values you already have. For this, you need to keep track of the signs, again: y=-4, r= squareroot(17)  [Note: r is always considered a positive number, whereas x and y can be either positive or negative, depending on where your diagrammed triangle is drawn]. You do the math, and express the final result in whatever format your course requires (i.e. if it says, no radicals in the denominator, then you have to multiply both numerator and denominator by squareroot(17) in order to clear the radical out of the denominator).

 

Key concepts: always draw a right triangle, starting from the origin, to visualize your "angle". And always keep your signs in mind when using your side lengths -- they're always "measured" from the appropriate reference axis (or from the origin, for the hypotenuse).