TRIGONOMETRY: DOUBLE-ANGLE FORMULAS

The above identities immediately follow from the sum formulas, as shown below.

sin2x = sin(x+x) = sinxcosx + cosxsinx = 2sinxcosx

cos2x = cos(x+x) = cosxcosx – sinxsinx = cos2x – sin2x

Note that the formula for cosine has three alternate forms. The first is in terms of sine and cosine,

the second in terms of cosine alone, and the third in terms of sine alone.

The second form readily follows from the first form by applying the Pythagorean Identity sin2x+cos2x=1

which gives sin2x = 1 – cos2x.

Therefore,

cos2x = cos2x – sin2x = cos2x -(1-cos2x) = cos2x -1+cos2x = 2cos2x – 1

The third form also readily follows from the first form by applying the same Pythagorean Identity

sin2x+cos2x=1 which gives cos2x = 1 – sin2x.

Therefore,

cos2x = cos2x – sin2x = (1-sin2x) – sin2x = 1-sin2x-sin2x = 1 – 2sin2x

Example

If sinx = 12/13 and x lies in quadrant II, find the value of each of the following:

a. sin2x b. cos2x c. tan2x

Solution:

a. Since sin2x = 2sinxcosx, we need the values of sinx and cosx. sinx is already given and so it remains to find cosx.

Use the Pythagorean Identity sin2x + cos2x = 1 to find cosx.

In choosing between the two possible values (positive or negative), consider the quadrant where x lies,
which is quadrant II, therefore cosx should be negative.

Therefore, cosx = -5/13.

Substitute values into the formula for sin2x.

Try these problems

QUESTIONS

1.  If sinx = 3/5 and x lies in quadrant II, find the value of each of the following:

     a. sin2x      b. cos2x     c. tan2x




2.  If sinx = -5/7 and x lies in quadrant III, find the value of each of the following:

     a. sin2x     b. cos2x     c. tan2x

ANSWERS