If tan(x) = −2 and x is in quadrant IV, find the exact values of the expressions without solving for x.

Question

A) sin(2x)

B)cos(2x)

c) tan(2x)

Answer 1

Tangent is -2 in quadrant 4.

Tangent = opposite/adjacent.

In quadrant 4, x>0 and y<0

So the adjacent x=1 and  the opposite y=-2.

By pythagorean theorem the hypothesis is square-root(5)

 

sine= opposite/ hypotneuse

       = -2/square-root(5)

 

Cosine= adjacent/hypotneuse

            = 1/square-root(5)

 

That's just the beginning.

As suggested the double angle formulas are needed to change

the trig expressions in terms of sine and cosine.

For example, sin(2x)= 2 sinx cosx

 = 2 (-2/square-root(5))(1/square-root(5)) = -4/5

 

Please do the same for the other 2 exercises. Find double angle expansions for cos(2x) and tan(2x)

In terms of sine, cosine, and tangent. Then plug in the above values and calculate 

Answer 2

you should know that 1 + tan²x = sec²x.

 

This enables you to compute sec x, to which you must affix the + sign because secant (reciprocal of cosine) is positive in that quadrant. Therefore you now know cos x and you can therefore easily compute sin x (which will have a - sign affixed because of Quadrant IV).

 

Knowing sine and cosine of x, use of the double angle formulas will enable you to evaluate the sine, cosine & tangent ratios of 2x--as Mark M suggests.

 

This is simple, staightforward work that you must be able to do in any Trigonometry class.

 

Remember to check your answers (can use calculator).